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Theorem maducoeval2 22541

Description: An entry of the adjunct (cofactor) matrix. (Contributed by SO, 17-Jul-2018.)

**Hypotheses**
Ref	Expression
madufval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
madufval.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
madufval.j	⊢ 𝐽 = (𝑁 maAdju 𝑅)
madufval.b	⊢ 𝐵 = (Base‘𝐴)
madufval.o	⊢ 1 = (1_r‘𝑅)
madufval.z	⊢ 0 = (0_g‘𝑅)

**Assertion**
Ref	Expression
maducoeval2	⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))))

Distinct variable groups: 𝑘,𝑁,𝑙 𝑅,𝑘,𝑙 𝑘,𝑀,𝑙 𝑘,𝐼,𝑙 𝑘,𝐻,𝑙 𝐵,𝑘,𝑙 0 ,𝑘 1 ,𝑘 Allowed substitution hints: 𝐴(𝑘,𝑙) 𝐷(𝑘,𝑙) 1 (𝑙) 𝐽(𝑘,𝑙) 0 (𝑙)

Proof of Theorem maducoeval2 Dummy variables 𝑛 𝑟 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2818	. . . . . . . 8 ⊢ (𝑚 = ∅ → (𝑘 ∈ 𝑚 ↔ 𝑘 ∈ ∅))
2	1	ifbid 4552	. . . . . . 7 ⊢ (𝑚 = ∅ → if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
3	2	ifeq2d 4549	. . . . . 6 ⊢ (𝑚 = ∅ → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
4	3	mpoeq3dv 7499	. . . . 5 ⊢ (𝑚 = ∅ → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
5	4	fveq2d 6901	. . . 4 ⊢ (𝑚 = ∅ → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
6	5	eqeq2d 2739	. . 3 ⊢ (𝑚 = ∅ → ((𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
7		eleq2 2818	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑘 ∈ 𝑚 ↔ 𝑘 ∈ 𝑛))
8	7	ifbid 4552	. . . . . . 7 ⊢ (𝑚 = 𝑛 → if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
9	8	ifeq2d 4549	. . . . . 6 ⊢ (𝑚 = 𝑛 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
10	9	mpoeq3dv 7499	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
11	10	fveq2d 6901	. . . 4 ⊢ (𝑚 = 𝑛 → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
12	11	eqeq2d 2739	. . 3 ⊢ (𝑚 = 𝑛 → ((𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
13		eleq2 2818	. . . . . . . 8 ⊢ (𝑚 = (𝑛 ∪ {𝑟}) → (𝑘 ∈ 𝑚 ↔ 𝑘 ∈ (𝑛 ∪ {𝑟})))
14	13	ifbid 4552	. . . . . . 7 ⊢ (𝑚 = (𝑛 ∪ {𝑟}) → if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
15	14	ifeq2d 4549	. . . . . 6 ⊢ (𝑚 = (𝑛 ∪ {𝑟}) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
16	15	mpoeq3dv 7499	. . . . 5 ⊢ (𝑚 = (𝑛 ∪ {𝑟}) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
17	16	fveq2d 6901	. . . 4 ⊢ (𝑚 = (𝑛 ∪ {𝑟}) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
18	17	eqeq2d 2739	. . 3 ⊢ (𝑚 = (𝑛 ∪ {𝑟}) → ((𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
19		eleq2 2818	. . . . . . . 8 ⊢ (𝑚 = (𝑁 ∖ {𝐻}) → (𝑘 ∈ 𝑚 ↔ 𝑘 ∈ (𝑁 ∖ {𝐻})))
20	19	ifbid 4552	. . . . . . 7 ⊢ (𝑚 = (𝑁 ∖ {𝐻}) → if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
21	20	ifeq2d 4549	. . . . . 6 ⊢ (𝑚 = (𝑁 ∖ {𝐻}) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
22	21	mpoeq3dv 7499	. . . . 5 ⊢ (𝑚 = (𝑁 ∖ {𝐻}) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
23	22	fveq2d 6901	. . . 4 ⊢ (𝑚 = (𝑁 ∖ {𝐻}) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
24	23	eqeq2d 2739	. . 3 ⊢ (𝑚 = (𝑁 ∖ {𝐻}) → ((𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
25		madufval.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
26		madufval.d	. . . . . 6 ⊢ 𝐷 = (𝑁 maDet 𝑅)
27		madufval.j	. . . . . 6 ⊢ 𝐽 = (𝑁 maAdju 𝑅)
28		madufval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
29		madufval.o	. . . . . 6 ⊢ 1 = (1_r‘𝑅)
30		madufval.z	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
31	25, 26, 27, 28, 29, 30	maducoeval 22540	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
32	31	3adant1l 1174	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
33		noel 4331	. . . . . . . 8 ⊢ ¬ 𝑘 ∈ ∅
34		iffalse 4538	. . . . . . . 8 ⊢ (¬ 𝑘 ∈ ∅ → if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = (𝑘𝑀𝑙))
35	33, 34	mp1i 13	. . . . . . 7 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = (𝑘𝑀𝑙))
36	35	ifeq2d 4549	. . . . . 6 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))
37	36	mpoeq3ia 7498	. . . . 5 ⊢ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))
38	37	fveq2i 6900	. . . 4 ⊢ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))))
39	32, 38	eqtr4di 2786	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
40		eqid 2728	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
41		eqid 2728	. . . . . . 7 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
42		eqid 2728	. . . . . . 7 ⊢ (._r‘𝑅) = (._r‘𝑅)
43		simpl1l 1222	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑅 ∈ CRing)
44		simp1r 1196	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → 𝑀 ∈ 𝐵)
45	25, 28	matrcl 22311	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
46	45	simpld 494	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
47	44, 46	syl 17	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → 𝑁 ∈ Fin)
48	47	adantr 480	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑁 ∈ Fin)
49		simp1l 1195	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → 𝑅 ∈ CRing)
50	49	ad2antrr 725	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 𝑅 ∈ CRing)
51		crngring 20184	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
52	50, 51	syl 17	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 𝑅 ∈ Ring)
53	40, 30	ring0cl 20202	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
54	52, 53	syl 17	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 0 ∈ (Base‘𝑅))
55		simpl1r 1223	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑀 ∈ 𝐵)
56	25, 40, 28	matbas2i 22323	. . . . . . . . . . 11 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
57		elmapi 8867	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
58	55, 56, 57	3syl 18	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
59	58	adantr 480	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
60		eldifi 4125	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛) → 𝑟 ∈ (𝑁 ∖ {𝐻}))
61	60	ad2antll 728	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑟 ∈ (𝑁 ∖ {𝐻}))
62	61	eldifad 3959	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑟 ∈ 𝑁)
63	62	adantr 480	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 𝑟 ∈ 𝑁)
64		simpr 484	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 𝑙 ∈ 𝑁)
65	59, 63, 64	fovcdmd 7593	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → (𝑟𝑀𝑙) ∈ (Base‘𝑅))
66	54, 65	ifcld 4575	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)) ∈ (Base‘𝑅))
67	40, 29	ringidcl 20201	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
68	52, 67	syl 17	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 1 ∈ (Base‘𝑅))
69	68, 54	ifcld 4575	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, 1 , 0 ) ∈ (Base‘𝑅))
70	54	3adant2 1129	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 0 ∈ (Base‘𝑅))
71	58	fovcdmda 7592	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ (𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁)) → (𝑘𝑀𝑙) ∈ (Base‘𝑅))
72	71	3impb 1113	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑘𝑀𝑙) ∈ (Base‘𝑅))
73	70, 72	ifcld 4575	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) ∈ (Base‘𝑅))
74	73, 72	ifcld 4575	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) ∈ (Base‘𝑅))
75		simpl2 1190	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝐼 ∈ 𝑁)
76	58, 62, 75	fovcdmd 7593	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝑟𝑀𝐼) ∈ (Base‘𝑅))
77		simpl3 1191	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝐻 ∈ 𝑁)
78		eldifsni 4794	. . . . . . . 8 ⊢ (𝑟 ∈ (𝑁 ∖ {𝐻}) → 𝑟 ≠ 𝐻)
79	61, 78	syl 17	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑟 ≠ 𝐻)
80	26, 40, 41, 42, 43, 48, 66, 69, 74, 76, 62, 77, 79	mdetero 22511	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
81		ifnot 4581	. . . . . . . . . . . . . . . . 17 ⊢ if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ) = if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))
82	81	eqcomi 2737	. . . . . . . . . . . . . . . 16 ⊢ if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)) = if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )
83	82	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)) = if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ))
84		ovif2 7519	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 )) = if(𝑙 = 𝐼, ((𝑟𝑀𝐼)(._r‘𝑅) 1 ), ((𝑟𝑀𝐼)(._r‘𝑅) 0 ))
85	76	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → (𝑟𝑀𝐼) ∈ (Base‘𝑅))
86	40, 42, 29	ringridm 20205	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Ring ∧ (𝑟𝑀𝐼) ∈ (Base‘𝑅)) → ((𝑟𝑀𝐼)(._r‘𝑅) 1 ) = (𝑟𝑀𝐼))
87	52, 85, 86	syl2anc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → ((𝑟𝑀𝐼)(._r‘𝑅) 1 ) = (𝑟𝑀𝐼))
88	87	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) ∧ 𝑙 = 𝐼) → ((𝑟𝑀𝐼)(._r‘𝑅) 1 ) = (𝑟𝑀𝐼))
89		oveq2 7428	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝐼 → (𝑟𝑀𝑙) = (𝑟𝑀𝐼))
90	89	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) ∧ 𝑙 = 𝐼) → (𝑟𝑀𝑙) = (𝑟𝑀𝐼))
91	88, 90	eqtr4d 2771	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) ∧ 𝑙 = 𝐼) → ((𝑟𝑀𝐼)(._r‘𝑅) 1 ) = (𝑟𝑀𝑙))
92	91	ifeq1da 4560	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, ((𝑟𝑀𝐼)(._r‘𝑅) 1 ), ((𝑟𝑀𝐼)(._r‘𝑅) 0 )) = if(𝑙 = 𝐼, (𝑟𝑀𝑙), ((𝑟𝑀𝐼)(._r‘𝑅) 0 )))
93	40, 42, 30	ringrz 20229	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ (𝑟𝑀𝐼) ∈ (Base‘𝑅)) → ((𝑟𝑀𝐼)(._r‘𝑅) 0 ) = 0 )
94	52, 85, 93	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → ((𝑟𝑀𝐼)(._r‘𝑅) 0 ) = 0 )
95	94	ifeq2d 4549	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, (𝑟𝑀𝑙), ((𝑟𝑀𝐼)(._r‘𝑅) 0 )) = if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ))
96	92, 95	eqtrd 2768	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, ((𝑟𝑀𝐼)(._r‘𝑅) 1 ), ((𝑟𝑀𝐼)(._r‘𝑅) 0 )) = if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ))
97	84, 96	eqtrid 2780	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → ((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 )) = if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ))
98	83, 97	oveq12d 7438	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )(+_g‘𝑅)if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )))
99		ringmnd 20182	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
100	52, 99	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 𝑅 ∈ Mnd)
101		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑙 = 𝐼 → ¬ 𝑙 = 𝐼)
102		imnan 399	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑙 = 𝐼 → ¬ 𝑙 = 𝐼) ↔ ¬ (¬ 𝑙 = 𝐼 ∧ 𝑙 = 𝐼))
103	101, 102	mpbi 229	. . . . . . . . . . . . . . . 16 ⊢ ¬ (¬ 𝑙 = 𝐼 ∧ 𝑙 = 𝐼)
104	103	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → ¬ (¬ 𝑙 = 𝐼 ∧ 𝑙 = 𝐼))
105	40, 30, 41	mndifsplit 22537	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Mnd ∧ (𝑟𝑀𝑙) ∈ (Base‘𝑅) ∧ ¬ (¬ 𝑙 = 𝐼 ∧ 𝑙 = 𝐼)) → if((¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼), (𝑟𝑀𝑙), 0 ) = (if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )(+_g‘𝑅)if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )))
106	100, 65, 104, 105	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if((¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼), (𝑟𝑀𝑙), 0 ) = (if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )(+_g‘𝑅)if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )))
107		pm2.1 895	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼)
108		iftrue 4535	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼) → if((¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼), (𝑟𝑀𝑙), 0 ) = (𝑟𝑀𝑙))
109	107, 108	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if((¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼), (𝑟𝑀𝑙), 0 ) = (𝑟𝑀𝑙))
110	98, 106, 109	3eqtr2d 2774	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑟𝑀𝑙))
111	110	3adant2 1129	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑟𝑀𝑙))
112		oveq1 7427	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑟 → (𝑘𝑀𝑙) = (𝑟𝑀𝑙))
113	112	eqeq2d 2739	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑟 → ((if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑘𝑀𝑙) ↔ (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑟𝑀𝑙)))
114	111, 113	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑘 = 𝑟 → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑘𝑀𝑙)))
115	114	imp 406	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑘𝑀𝑙))
116		iftrue 4535	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑟 → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))))
117	116	adantl 481	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))))
118	79	neneqd 2942	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → ¬ 𝑟 = 𝐻)
119	118	3ad2ant1 1131	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → ¬ 𝑟 = 𝐻)
120		eqeq1 2732	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑟 → (𝑘 = 𝐻 ↔ 𝑟 = 𝐻))
121	120	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑟 → (¬ 𝑘 = 𝐻 ↔ ¬ 𝑟 = 𝐻))
122	119, 121	syl5ibrcom 246	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑘 = 𝑟 → ¬ 𝑘 = 𝐻))
123	122	imp 406	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → ¬ 𝑘 = 𝐻)
124	123	iffalsed 4540	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
125		eldifn 4126	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛) → ¬ 𝑟 ∈ 𝑛)
126	125	ad2antll 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → ¬ 𝑟 ∈ 𝑛)
127	126	3ad2ant1 1131	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → ¬ 𝑟 ∈ 𝑛)
128		eleq1w 2812	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑟 → (𝑘 ∈ 𝑛 ↔ 𝑟 ∈ 𝑛))
129	128	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑟 → (¬ 𝑘 ∈ 𝑛 ↔ ¬ 𝑟 ∈ 𝑛))
130	127, 129	syl5ibrcom 246	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑘 = 𝑟 → ¬ 𝑘 ∈ 𝑛))
131	130	imp 406	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → ¬ 𝑘 ∈ 𝑛)
132	131	iffalsed 4540	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = (𝑘𝑀𝑙))
133	124, 132	eqtrd 2768	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = (𝑘𝑀𝑙))
134	115, 117, 133	3eqtr4d 2778	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
135		iffalse 4538	. . . . . . . . . 10 ⊢ (¬ 𝑘 = 𝑟 → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
136	135	adantl 481	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝑟) → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
137	134, 136	pm2.61dan 812	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
138	137	mpoeq3dva 7497	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
139	138	fveq2d 6901	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
140		neeq2 3001	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐻 → (𝑟 ≠ 𝑘 ↔ 𝑟 ≠ 𝐻))
141	140	biimparc 479	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → 𝑟 ≠ 𝑘)
142	141	necomd 2993	. . . . . . . . . . . . 13 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → 𝑘 ≠ 𝑟)
143	142	neneqd 2942	. . . . . . . . . . . 12 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → ¬ 𝑘 = 𝑟)
144	143	iffalsed 4540	. . . . . . . . . . 11 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑙 = 𝐼, 1 , 0 )) = if(𝑙 = 𝐼, 1 , 0 ))
145		iftrue 4535	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 1 , 0 ))
146	145	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 1 , 0 ))
147	146	ifeq2d 4549	. . . . . . . . . . 11 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑙 = 𝐼, 1 , 0 )))
148		iftrue 4535	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 1 , 0 ))
149	148	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 1 , 0 ))
150	144, 147, 149	3eqtr4d 2778	. . . . . . . . . 10 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
151	112	ifeq2d 4549	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑟 → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)))
152		vsnid 4666	. . . . . . . . . . . . . . . . 17 ⊢ 𝑟 ∈ {𝑟}
153		elun2 4177	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ {𝑟} → 𝑟 ∈ (𝑛 ∪ {𝑟}))
154	152, 153	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ 𝑟 ∈ (𝑛 ∪ {𝑟})
155		eleq1w 2812	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑟 → (𝑘 ∈ (𝑛 ∪ {𝑟}) ↔ 𝑟 ∈ (𝑛 ∪ {𝑟})))
156	154, 155	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑟 → 𝑘 ∈ (𝑛 ∪ {𝑟}))
157	156	iftrued 4537	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑟 → if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)))
158		iftrue 4535	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑟 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)))
159	151, 157, 158	3eqtr4rd 2779	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑟 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
160	159	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
161		iffalse 4538	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑘 = 𝑟 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
162		orc 866	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑛 → (𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟))
163		orel2 889	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑘 = 𝑟 → ((𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟) → 𝑘 ∈ 𝑛))
164	162, 163	impbid2 225	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 𝑟 → (𝑘 ∈ 𝑛 ↔ (𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟)))
165		elun 4147	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑛 ∪ {𝑟}) ↔ (𝑘 ∈ 𝑛 ∨ 𝑘 ∈ {𝑟}))
166		velsn 4645	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ {𝑟} ↔ 𝑘 = 𝑟)
167	166	orbi2i 911	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ 𝑛 ∨ 𝑘 ∈ {𝑟}) ↔ (𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟))
168	165, 167	bitr2i 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟) ↔ 𝑘 ∈ (𝑛 ∪ {𝑟}))
169	164, 168	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑘 = 𝑟 → (𝑘 ∈ 𝑛 ↔ 𝑘 ∈ (𝑛 ∪ {𝑟})))
170	169	ifbid 4552	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑘 = 𝑟 → if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
171	161, 170	eqtrd 2768	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 = 𝑟 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
172	171	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻) ∧ ¬ 𝑘 = 𝑟) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
173	160, 172	pm2.61dan 812	. . . . . . . . . . 11 ⊢ ((𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
174		iffalse 4538	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
175	174	ifeq2d 4549	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = 𝐻 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
176	175	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
177		iffalse 4538	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
178	177	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
179	173, 176, 178	3eqtr4d 2778	. . . . . . . . . 10 ⊢ ((𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
180	150, 179	pm2.61dan 812	. . . . . . . . 9 ⊢ (𝑟 ≠ 𝐻 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
181	180	mpoeq3dv 7499	. . . . . . . 8 ⊢ (𝑟 ≠ 𝐻 → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
182	181	fveq2d 6901	. . . . . . 7 ⊢ (𝑟 ≠ 𝐻 → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
183	79, 182	syl 17	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
184	80, 139, 183	3eqtr3d 2776	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
185	184	eqeq2d 2739	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → ((𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
186	185	biimpd 228	. . 3 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → ((𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
187		difss 4130	. . . 4 ⊢ (𝑁 ∖ {𝐻}) ⊆ 𝑁
188		ssfi 9197	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ (𝑁 ∖ {𝐻}) ⊆ 𝑁) → (𝑁 ∖ {𝐻}) ∈ Fin)
189	47, 187, 188	sylancl 585	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝑁 ∖ {𝐻}) ∈ Fin)
190	6, 12, 18, 24, 39, 186, 189	findcard2d 9190	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
191		iba 527	. . . . . . . 8 ⊢ (𝑘 = 𝐻 → (𝑙 = 𝐼 ↔ (𝑙 = 𝐼 ∧ 𝑘 = 𝐻)))
192	191	ifbid 4552	. . . . . . 7 ⊢ (𝑘 = 𝐻 → if(𝑙 = 𝐼, 1 , 0 ) = if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ))
193		iftrue 4535	. . . . . . 7 ⊢ (𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 1 , 0 ))
194		iftrue 4535	. . . . . . . 8 ⊢ ((𝑘 = 𝐻 ∨ 𝑙 = 𝐼) → if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)) = if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ))
195	194	orcs 874	. . . . . . 7 ⊢ (𝑘 = 𝐻 → if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)) = if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ))
196	192, 193, 195	3eqtr4d 2778	. . . . . 6 ⊢ (𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
197	196	adantl 481	. . . . 5 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
198		iffalse 4538	. . . . . . 7 ⊢ (¬ 𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
199	198	adantl 481	. . . . . 6 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
200		neqne 2945	. . . . . . . . . 10 ⊢ (¬ 𝑘 = 𝐻 → 𝑘 ≠ 𝐻)
201	200	anim2i 616	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝑁 ∧ ¬ 𝑘 = 𝐻) → (𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐻))
202	201	adantlr 714	. . . . . . . 8 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐻) → (𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐻))
203		eldifsn 4791	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑁 ∖ {𝐻}) ↔ (𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐻))
204	202, 203	sylibr 233	. . . . . . 7 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐻) → 𝑘 ∈ (𝑁 ∖ {𝐻}))
205	204	iftrued 4537	. . . . . 6 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐻) → if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)))
206		biorf 935	. . . . . . . 8 ⊢ (¬ 𝑘 = 𝐻 → (𝑙 = 𝐼 ↔ (𝑘 = 𝐻 ∨ 𝑙 = 𝐼)))
207		id 22	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = 𝐻 → ¬ 𝑘 = 𝐻)
208	207	intnand 488	. . . . . . . . . 10 ⊢ (¬ 𝑘 = 𝐻 → ¬ (𝑙 = 𝐼 ∧ 𝑘 = 𝐻))
209	208	iffalsed 4540	. . . . . . . . 9 ⊢ (¬ 𝑘 = 𝐻 → if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ) = 0 )
210	209	eqcomd 2734	. . . . . . . 8 ⊢ (¬ 𝑘 = 𝐻 → 0 = if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ))
211	206, 210	ifbieq1d 4553	. . . . . . 7 ⊢ (¬ 𝑘 = 𝐻 → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) = if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
212	211	adantl 481	. . . . . 6 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐻) → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) = if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
213	199, 205, 212	3eqtrd 2772	. . . . 5 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
214	197, 213	pm2.61dan 812	. . . 4 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
215	214	mpoeq3ia 7498	. . 3 ⊢ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
216	215	fveq2i 6900	. 2 ⊢ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙))))
217	190, 216	eqtrdi 2784	1 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 846 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 ∖ cdif 3944 ∪ cun 3945 ⊆ wss 3947 ∅c0 4323 ifcif 4529 {csn 4629 × cxp 5676 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 ↑_m cmap 8844 Fincfn 8963 Basecbs 17179 +_gcplusg 17232 ._rcmulr 17233 0_gc0g 17420 Mndcmnd 18693 1_rcur 20120 Ringcrg 20172 CRingccrg 20173 Mat cmat 22306 maDet cmdat 22485 maAdju cmadu 22533

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-addf 11217 ax-mulf 11218

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-xor 1506 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8846 df-pm 8847 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-xnn0 12575 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-word 14497 df-lsw 14545 df-concat 14553 df-s1 14578 df-substr 14623 df-pfx 14653 df-splice 14732 df-reverse 14741 df-s2 14831 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-efmnd 18820 df-grp 18892 df-minusg 18893 df-mulg 19023 df-subg 19077 df-ghm 19167 df-gim 19212 df-cntz 19267 df-oppg 19296 df-symg 19321 df-pmtr 19396 df-psgn 19445 df-evpm 19446 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-rhm 20410 df-subrng 20482 df-subrg 20507 df-drng 20625 df-sra 21057 df-rgmod 21058 df-cnfld 21279 df-zring 21372 df-zrh 21428 df-dsmm 21665 df-frlm 21680 df-mat 22307 df-mdet 22486 df-madu 22535

This theorem is referenced by: madutpos 22543

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