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Theorem mvmulfval 22464

Description: Functional value of the matrix vector multiplication operator. (Contributed by AV, 23-Feb-2019.)

**Hypotheses**
Ref	Expression
mvmulfval.x	⊢ × = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)
mvmulfval.b	⊢ 𝐵 = (Base‘𝑅)
mvmulfval.t	⊢ · = (._r‘𝑅)
mvmulfval.r	⊢ (𝜑 → 𝑅 ∈ 𝑉)
mvmulfval.m	⊢ (𝜑 → 𝑀 ∈ Fin)
mvmulfval.n	⊢ (𝜑 → 𝑁 ∈ Fin)

**Assertion**
Ref	Expression
mvmulfval	⊢ (𝜑 → × = (𝑥 ∈ (𝐵 ↑_m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑_m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))))

Distinct variable groups: 𝑖,𝑗,𝑥,𝑦,𝜑 𝑖,𝑀,𝑗,𝑥,𝑦 𝑖,𝑁,𝑗,𝑥,𝑦 𝑅,𝑖,𝑗,𝑥,𝑦 𝑥,𝐵,𝑦 𝑥, · ,𝑦,𝑖 Allowed substitution hints: 𝐵(𝑖,𝑗) · (𝑗) × (𝑥,𝑦,𝑖,𝑗) 𝑉(𝑥,𝑦,𝑖,𝑗)

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

Step	Hyp	Ref	Expression
1		mvmulfval.x	. 2 ⊢ × = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)
2		df-mvmul 22463	. . . 4 ⊢ maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ ⦋(1^st ‘𝑜) / 𝑚⦌⦋(2^nd ‘𝑜) / 𝑛⦌(𝑥 ∈ ((Base‘𝑟) ↑_m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑_m 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σ_g (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗)))))))
3	2	a1i 11	. . 3 ⊢ (𝜑 → maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ ⦋(1^st ‘𝑜) / 𝑚⦌⦋(2^nd ‘𝑜) / 𝑛⦌(𝑥 ∈ ((Base‘𝑟) ↑_m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑_m 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σ_g (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗))))))))
4		fvex 6915	. . . . 5 ⊢ (1^st ‘𝑜) ∈ V
5		fvex 6915	. . . . 5 ⊢ (2^nd ‘𝑜) ∈ V
6		xpeq12 5707	. . . . . . 7 ⊢ ((𝑚 = (1^st ‘𝑜) ∧ 𝑛 = (2^nd ‘𝑜)) → (𝑚 × 𝑛) = ((1^st ‘𝑜) × (2^nd ‘𝑜)))
7	6	oveq2d 7442	. . . . . 6 ⊢ ((𝑚 = (1^st ‘𝑜) ∧ 𝑛 = (2^nd ‘𝑜)) → ((Base‘𝑟) ↑_m (𝑚 × 𝑛)) = ((Base‘𝑟) ↑_m ((1^st ‘𝑜) × (2^nd ‘𝑜))))
8		oveq2 7434	. . . . . . 7 ⊢ (𝑛 = (2^nd ‘𝑜) → ((Base‘𝑟) ↑_m 𝑛) = ((Base‘𝑟) ↑_m (2^nd ‘𝑜)))
9	8	adantl 480	. . . . . 6 ⊢ ((𝑚 = (1^st ‘𝑜) ∧ 𝑛 = (2^nd ‘𝑜)) → ((Base‘𝑟) ↑_m 𝑛) = ((Base‘𝑟) ↑_m (2^nd ‘𝑜)))
10		simpl 481	. . . . . . 7 ⊢ ((𝑚 = (1^st ‘𝑜) ∧ 𝑛 = (2^nd ‘𝑜)) → 𝑚 = (1^st ‘𝑜))
11		simpr 483	. . . . . . . . 9 ⊢ ((𝑚 = (1^st ‘𝑜) ∧ 𝑛 = (2^nd ‘𝑜)) → 𝑛 = (2^nd ‘𝑜))
12	11	mpteq1d 5247	. . . . . . . 8 ⊢ ((𝑚 = (1^st ‘𝑜) ∧ 𝑛 = (2^nd ‘𝑜)) → (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗))) = (𝑗 ∈ (2^nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗))))
13	12	oveq2d 7442	. . . . . . 7 ⊢ ((𝑚 = (1^st ‘𝑜) ∧ 𝑛 = (2^nd ‘𝑜)) → (𝑟 Σ_g (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗)))) = (𝑟 Σ_g (𝑗 ∈ (2^nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗)))))
14	10, 13	mpteq12dv 5243	. . . . . 6 ⊢ ((𝑚 = (1^st ‘𝑜) ∧ 𝑛 = (2^nd ‘𝑜)) → (𝑖 ∈ 𝑚 ↦ (𝑟 Σ_g (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗))))) = (𝑖 ∈ (1^st ‘𝑜) ↦ (𝑟 Σ_g (𝑗 ∈ (2^nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗))))))
15	7, 9, 14	mpoeq123dv 7501	. . . . 5 ⊢ ((𝑚 = (1^st ‘𝑜) ∧ 𝑛 = (2^nd ‘𝑜)) → (𝑥 ∈ ((Base‘𝑟) ↑_m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑_m 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σ_g (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗)))))) = (𝑥 ∈ ((Base‘𝑟) ↑_m ((1^st ‘𝑜) × (2^nd ‘𝑜))), 𝑦 ∈ ((Base‘𝑟) ↑_m (2^nd ‘𝑜)) ↦ (𝑖 ∈ (1^st ‘𝑜) ↦ (𝑟 Σ_g (𝑗 ∈ (2^nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗)))))))
16	4, 5, 15	csbie2 3936	. . . 4 ⊢ ⦋(1^st ‘𝑜) / 𝑚⦌⦋(2^nd ‘𝑜) / 𝑛⦌(𝑥 ∈ ((Base‘𝑟) ↑_m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑_m 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σ_g (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗)))))) = (𝑥 ∈ ((Base‘𝑟) ↑_m ((1^st ‘𝑜) × (2^nd ‘𝑜))), 𝑦 ∈ ((Base‘𝑟) ↑_m (2^nd ‘𝑜)) ↦ (𝑖 ∈ (1^st ‘𝑜) ↦ (𝑟 Σ_g (𝑗 ∈ (2^nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗))))))
17		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → 𝑟 = 𝑅)
18	17	fveq2d 6906	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = (Base‘𝑅))
19		mvmulfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
20	18, 19	eqtr4di 2786	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = 𝐵)
21		fveq2 6902	. . . . . . . . 9 ⊢ (𝑜 = ⟨𝑀, 𝑁⟩ → (1^st ‘𝑜) = (1^st ‘⟨𝑀, 𝑁⟩))
22	21	ad2antll 727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (1^st ‘𝑜) = (1^st ‘⟨𝑀, 𝑁⟩))
23		mvmulfval.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ Fin)
24		mvmulfval.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ Fin)
25		op1stg 8011	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (1^st ‘⟨𝑀, 𝑁⟩) = 𝑀)
26	23, 24, 25	syl2anc 582	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘⟨𝑀, 𝑁⟩) = 𝑀)
27	26	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (1^st ‘⟨𝑀, 𝑁⟩) = 𝑀)
28	22, 27	eqtrd 2768	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (1^st ‘𝑜) = 𝑀)
29		fveq2 6902	. . . . . . . . 9 ⊢ (𝑜 = ⟨𝑀, 𝑁⟩ → (2^nd ‘𝑜) = (2^nd ‘⟨𝑀, 𝑁⟩))
30	29	ad2antll 727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (2^nd ‘𝑜) = (2^nd ‘⟨𝑀, 𝑁⟩))
31		op2ndg 8012	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (2^nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
32	23, 24, 31	syl2anc 582	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
33	32	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (2^nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
34	30, 33	eqtrd 2768	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (2^nd ‘𝑜) = 𝑁)
35	28, 34	xpeq12d 5713	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((1^st ‘𝑜) × (2^nd ‘𝑜)) = (𝑀 × 𝑁))
36	20, 35	oveq12d 7444	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑_m ((1^st ‘𝑜) × (2^nd ‘𝑜))) = (𝐵 ↑_m (𝑀 × 𝑁)))
37	20, 34	oveq12d 7444	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑_m (2^nd ‘𝑜)) = (𝐵 ↑_m 𝑁))
38		fveq2 6902	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (._r‘𝑟) = (._r‘𝑅))
39	38	adantr 479	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩) → (._r‘𝑟) = (._r‘𝑅))
40	39	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (._r‘𝑟) = (._r‘𝑅))
41		mvmulfval.t	. . . . . . . . . 10 ⊢ · = (._r‘𝑅)
42	40, 41	eqtr4di 2786	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (._r‘𝑟) = · )
43	42	oveqd 7443	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗)) = ((𝑖𝑥𝑗) · (𝑦‘𝑗)))
44	34, 43	mpteq12dv 5243	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑗 ∈ (2^nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))))
45	17, 44	oveq12d 7444	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑟 Σ_g (𝑗 ∈ (2^nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗)))) = (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))
46	28, 45	mpteq12dv 5243	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑖 ∈ (1^st ‘𝑜) ↦ (𝑟 Σ_g (𝑗 ∈ (2^nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗))))) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))))))
47	36, 37, 46	mpoeq123dv 7501	. . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑥 ∈ ((Base‘𝑟) ↑_m ((1^st ‘𝑜) × (2^nd ‘𝑜))), 𝑦 ∈ ((Base‘𝑟) ↑_m (2^nd ‘𝑜)) ↦ (𝑖 ∈ (1^st ‘𝑜) ↦ (𝑟 Σ_g (𝑗 ∈ (2^nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗)))))) = (𝑥 ∈ (𝐵 ↑_m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑_m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))))
48	16, 47	eqtrid 2780	. . 3 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ⦋(1^st ‘𝑜) / 𝑚⦌⦋(2^nd ‘𝑜) / 𝑛⦌(𝑥 ∈ ((Base‘𝑟) ↑_m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑_m 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σ_g (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗)))))) = (𝑥 ∈ (𝐵 ↑_m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑_m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))))
49		mvmulfval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
50	49	elexd 3494	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
51		opex 5470	. . . 4 ⊢ ⟨𝑀, 𝑁⟩ ∈ V
52	51	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ V)
53		ovex 7459	. . . . 5 ⊢ (𝐵 ↑_m (𝑀 × 𝑁)) ∈ V
54		ovex 7459	. . . . 5 ⊢ (𝐵 ↑_m 𝑁) ∈ V
55	53, 54	mpoex 8090	. . . 4 ⊢ (𝑥 ∈ (𝐵 ↑_m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑_m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))) ∈ V
56	55	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐵 ↑_m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑_m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))) ∈ V)
57	3, 48, 50, 52, 56	ovmpod 7579	. 2 ⊢ (𝜑 → (𝑅 maVecMul ⟨𝑀, 𝑁⟩) = (𝑥 ∈ (𝐵 ↑_m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑_m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))))
58	1, 57	eqtrid 2780	1 ⊢ (𝜑 → × = (𝑥 ∈ (𝐵 ↑_m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑_m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ⦋csb 3894 ⟨cop 4638 ↦ cmpt 5235 × cxp 5680 ‘cfv 6553 (class class class)co 7426 ∈ cmpo 7428 1^st c1st 7997 2^nd c2nd 7998 ↑_m cmap 8851 Fincfn 8970 Basecbs 17187 ._rcmulr 17241 Σ_g cgsu 17429 maVecMul cmvmul 22462

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-mvmul 22463

This theorem is referenced by: mvmulval 22465 mavmuldm 22472 mavmul0g 22475

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