Theorem evlseu 22098

Description: For a given interpretation of the variables 𝐺 and of the scalars 𝐹, this extends to a homomorphic interpretation of the polynomial ring in exactly one way. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 11-Apr-2024.)

Hypotheses

Ref	Expression
evlseu.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
evlseu.c	⊢ 𝐶 = (Base‘𝑆)
evlseu.a	⊢ 𝐴 = (algSc‘𝑃)
evlseu.v	⊢ 𝑉 = (𝐼 mVar 𝑅)
evlseu.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
evlseu.r	⊢ (𝜑 → 𝑅 ∈ CRing)
evlseu.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evlseu.f	⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))
evlseu.g	⊢ (𝜑 → 𝐺:𝐼⟶𝐶)

Assertion

Ref	Expression
evlseu	⊢ (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))

Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝑚,𝐺   𝑚,𝐼   𝑃,𝑚   𝜑,𝑚   𝑆,𝑚   𝑚,𝑉

Allowed substitution hints:   𝐶(𝑚)   𝑅(𝑚)   𝑊(𝑚)

Proof of Theorem evlseu

Dummy variables 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlseu.p	. . . 4 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
2		eqid 2726	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
3		evlseu.c	. . . 4 ⊢ 𝐶 = (Base‘𝑆)
4		eqid 2726	. . . 4 ⊢ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} = {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin}
5		eqid 2726	. . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
6		eqid 2726	. . . 4 ⊢ (.g‘(mulGrp‘𝑆)) = (.g‘(mulGrp‘𝑆))
7		eqid 2726	. . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆)
8		evlseu.v	. . . 4 ⊢ 𝑉 = (𝐼 mVar 𝑅)
9		eqid 2726	. . . 4 ⊢ (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺))))))
10		evlseu.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
11		evlseu.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing)
12		evlseu.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ CRing)
13		evlseu.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))
14		evlseu.g	. . . 4 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶)
15		evlseu.a	. . . 4 ⊢ 𝐴 = (algSc‘𝑃)
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	evlslem1 22097	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∈ (𝑃 RingHom 𝑆) ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺))
17		coeq1 5864	. . . . . . 7 ⊢ (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) → (𝑚 ∘ 𝐴) = ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴))
18	17	eqeq1d 2728	. . . . . 6 ⊢ (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) → ((𝑚 ∘ 𝐴) = 𝐹 ↔ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹))
19		coeq1 5864	. . . . . . 7 ⊢ (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) → (𝑚 ∘ 𝑉) = ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉))
20	19	eqeq1d 2728	. . . . . 6 ⊢ (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) → ((𝑚 ∘ 𝑉) = 𝐺 ↔ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺))
21	18, 20	anbi12d 630	. . . . 5 ⊢ (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) → (((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) ↔ (((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺)))
22	21	rspcev 3608	. . . 4 ⊢ (((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∈ (𝑃 RingHom 𝑆) ∧ (((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺)) → ∃𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))
23	22	3impb 1112	. . 3 ⊢ (((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∈ (𝑃 RingHom 𝑆) ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(.r‘𝑆)((mulGrp‘𝑆) Σg (𝑦 ∘f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺) → ∃𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))
24	16, 23	syl 17	. 2 ⊢ (𝜑 → ∃𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))
25		eqid 2726	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
26		crngring 20228	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
27	11, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
28	1, 2, 25, 15, 10, 27	mplasclf 22078	. . . . . . . . 9 ⊢ (𝜑 → 𝐴:(Base‘𝑅)⟶(Base‘𝑃))
29	28	ffund 6732	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐴)
30		funcoeqres 6874	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ (𝑚 ∘ 𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹 ∘ ◡𝐴))
31	29, 30	sylan 578	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∘ 𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹 ∘ ◡𝐴))
32	1, 8, 2, 10, 27	mvrf2 22002	. . . . . . . . 9 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑃))
33	32	ffund 6732	. . . . . . . 8 ⊢ (𝜑 → Fun 𝑉)
34		funcoeqres 6874	. . . . . . . 8 ⊢ ((Fun 𝑉 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ◡𝑉))
35	33, 34	sylan 578	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ◡𝑉))
36	31, 35	anim12dan 617	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺)) → ((𝑚 ↾ ran 𝐴) = (𝐹 ∘ ◡𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ◡𝑉)))
37	36	ex 411	. . . . 5 ⊢ (𝜑 → (((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) → ((𝑚 ↾ ran 𝐴) = (𝐹 ∘ ◡𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ◡𝑉))))
38		resundi 6003	. . . . . 6 ⊢ (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉))
39		uneq12 4158	. . . . . 6 ⊢ (((𝑚 ↾ ran 𝐴) = (𝐹 ∘ ◡𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ◡𝑉)) → ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉)))
40	38, 39	eqtrid 2778	. . . . 5 ⊢ (((𝑚 ↾ ran 𝐴) = (𝐹 ∘ ◡𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ◡𝑉)) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉)))
41	37, 40	syl6 35	. . . 4 ⊢ (𝜑 → (((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉))))
42	41	ralrimivw 3140	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ (𝑃 RingHom 𝑆)(((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉))))
43		eqtr3 2752	. . . . . 6 ⊢ (((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉)) ∧ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉))) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)))
44		eqid 2726	. . . . . . . . . . . . 13 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
45	44, 10, 11	psrassa 21982	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg)
46		eqid 2726	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
47	44, 8, 46, 10, 27	mvrf 21994	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅)))
48	47	frnd 6736	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
49		eqid 2726	. . . . . . . . . . . . 13 ⊢ (AlgSpan‘(𝐼 mPwSer 𝑅)) = (AlgSpan‘(𝐼 mPwSer 𝑅))
50		eqid 2726	. . . . . . . . . . . . 13 ⊢ (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘(𝐼 mPwSer 𝑅))
51		eqid 2726	. . . . . . . . . . . . 13 ⊢ (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅))) = (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))
52	49, 50, 51, 46	aspval2 21895	. . . . . . . . . . . 12 ⊢ (((𝐼 mPwSer 𝑅) ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘(𝐼 mPwSer 𝑅))) → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)))
53	45, 48, 52	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)))
54	1, 44, 8, 49, 10, 11	mplbas2 22049	. . . . . . . . . . 11 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑃))
55	44, 1, 2, 10, 27	mplsubrg 22014	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
56	1, 44, 2	mplval2 22005	. . . . . . . . . . . . . . . 16 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑃))
57	56	subsubrg2 20583	. . . . . . . . . . . . . . 15 ⊢ ((Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → (SubRing‘𝑃) = ((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
58	55, 57	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (SubRing‘𝑃) = ((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
59	58	fveq2d 6905	. . . . . . . . . . . . 13 ⊢ (𝜑 → (mrCls‘(SubRing‘𝑃)) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))))
60	50, 56	ressascl 21893	. . . . . . . . . . . . . . . . 17 ⊢ ((Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘𝑃))
61	55, 60	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘𝑃))
62	15, 61	eqtr4id 2785	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 = (algSc‘(𝐼 mPwSer 𝑅)))
63	62	rneqd 5944	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐴 = ran (algSc‘(𝐼 mPwSer 𝑅)))
64	63	uneq1d 4162	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran 𝐴 ∪ ran 𝑉) = (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉))
65	59, 64	fveq12d 6908	. . . . . . . . . . . 12 ⊢ (𝜑 → ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)) = ((mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)))
66		assaring 21859	. . . . . . . . . . . . . 14 ⊢ ((𝐼 mPwSer 𝑅) ∈ AssAlg → (𝐼 mPwSer 𝑅) ∈ Ring)
67	46	subrgmre 20581	. . . . . . . . . . . . . 14 ⊢ ((𝐼 mPwSer 𝑅) ∈ Ring → (SubRing‘(𝐼 mPwSer 𝑅)) ∈ (Moore‘(Base‘(𝐼 mPwSer 𝑅))))
68	45, 66, 67	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (SubRing‘(𝐼 mPwSer 𝑅)) ∈ (Moore‘(Base‘(𝐼 mPwSer 𝑅))))
69	28	frnd 6736	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐴 ⊆ (Base‘𝑃))
70	63, 69	eqsstrrd 4019	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran (algSc‘(𝐼 mPwSer 𝑅)) ⊆ (Base‘𝑃))
71	32	frnd 6736	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
72	70, 71	unssd 4187	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉) ⊆ (Base‘𝑃))
73		eqid 2726	. . . . . . . . . . . . . 14 ⊢ (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
74	51, 73	submrc 17641	. . . . . . . . . . . . 13 ⊢ (((SubRing‘(𝐼 mPwSer 𝑅)) ∈ (Moore‘(Base‘(𝐼 mPwSer 𝑅))) ∧ (Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) ∧ (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉) ⊆ (Base‘𝑃)) → ((mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)) = ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)))
75	68, 55, 72, 74	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (𝜑 → ((mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)) = ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)))
76	65, 75	eqtr2d 2767	. . . . . . . . . . 11 ⊢ (𝜑 → ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
77	53, 54, 76	3eqtr3d 2774	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑃) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
78	77	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → (Base‘𝑃) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
79	1, 10, 27	mplringd 22032	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ Ring)
80	2	subrgmre 20581	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ Ring → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
81	79, 80	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
82	81	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
83		simpr 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛))
84		rhmeql 20587	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆)) → dom (𝑚 ∩ 𝑛) ∈ (SubRing‘𝑃))
85	84	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → dom (𝑚 ∩ 𝑛) ∈ (SubRing‘𝑃))
86		eqid 2726	. . . . . . . . . . 11 ⊢ (mrCls‘(SubRing‘𝑃)) = (mrCls‘(SubRing‘𝑃))
87	86	mrcsscl 17633	. . . . . . . . . 10 ⊢ (((SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛) ∧ dom (𝑚 ∩ 𝑛) ∈ (SubRing‘𝑃)) → ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)) ⊆ dom (𝑚 ∩ 𝑛))
88	82, 83, 85, 87	syl3anc 1368	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)) ⊆ dom (𝑚 ∩ 𝑛))
89	78, 88	eqsstrd 4018	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛))
90	89	ex 411	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛) → (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
91		simprl 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 ∈ (𝑃 RingHom 𝑆))
92	2, 3	rhmf 20467	. . . . . . . . 9 ⊢ (𝑚 ∈ (𝑃 RingHom 𝑆) → 𝑚:(Base‘𝑃)⟶𝐶)
93		ffn 6728	. . . . . . . . 9 ⊢ (𝑚:(Base‘𝑃)⟶𝐶 → 𝑚 Fn (Base‘𝑃))
94	91, 92, 93	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 Fn (Base‘𝑃))
95		simprr 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 ∈ (𝑃 RingHom 𝑆))
96	2, 3	rhmf 20467	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑃 RingHom 𝑆) → 𝑛:(Base‘𝑃)⟶𝐶)
97		ffn 6728	. . . . . . . . 9 ⊢ (𝑛:(Base‘𝑃)⟶𝐶 → 𝑛 Fn (Base‘𝑃))
98	95, 96, 97	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 Fn (Base‘𝑃))
99	69	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝐴 ⊆ (Base‘𝑃))
100	71	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝑉 ⊆ (Base‘𝑃))
101	99, 100	unssd 4187	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (ran 𝐴 ∪ ran 𝑉) ⊆ (Base‘𝑃))
102		fnreseql 7061	. . . . . . . 8 ⊢ ((𝑚 Fn (Base‘𝑃) ∧ 𝑛 Fn (Base‘𝑃) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ (Base‘𝑃)) → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) ↔ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)))
103	94, 98, 101, 102	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) ↔ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)))
104		fneqeql2 7060	. . . . . . . 8 ⊢ ((𝑚 Fn (Base‘𝑃) ∧ 𝑛 Fn (Base‘𝑃)) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
105	94, 98, 104	syl2anc 582	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
106	90, 103, 105	3imtr4d 293	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) → 𝑚 = 𝑛))
107	43, 106	syl5 34	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉)) ∧ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉))) → 𝑚 = 𝑛))
108	107	ralrimivva 3191	. . . 4 ⊢ (𝜑 → ∀𝑚 ∈ (𝑃 RingHom 𝑆)∀𝑛 ∈ (𝑃 RingHom 𝑆)(((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉)) ∧ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉))) → 𝑚 = 𝑛))
109		reseq1 5983	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)))
110	109	eqeq1d 2728	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉)) ↔ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉))))
111	110	rmo4 3724	. . . 4 ⊢ (∃*𝑚 ∈ (𝑃 RingHom 𝑆)(𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉)) ↔ ∀𝑚 ∈ (𝑃 RingHom 𝑆)∀𝑛 ∈ (𝑃 RingHom 𝑆)(((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉)) ∧ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉))) → 𝑚 = 𝑛))
112	108, 111	sylibr 233	. . 3 ⊢ (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)(𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉)))
113		rmoim 3734	. . 3 ⊢ (∀𝑚 ∈ (𝑃 RingHom 𝑆)(((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉))) → (∃*𝑚 ∈ (𝑃 RingHom 𝑆)(𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉)) → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺)))
114	42, 112, 113	sylc 65	. 2 ⊢ (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))
115		reu5 3366	. 2 ⊢ (∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) ↔ (∃𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) ∧ ∃*𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺)))
116	24, 114, 115	sylanbrc 581	1 ⊢ (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  ∀wral 3051  ∃wrex 3060  ∃!wreu 3362  ∃*wrmo 3363  {crab 3419   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  𝒫 cpw 4607   ↦ cmpt 5236  ^◡ccnv 5681  dom cdm 5682  ran crn 5683   ↾ cres 5684   “ cima 5685   ∘ ccom 5686  Fun wfun 6548   Fn wfn 6549  ⟶wf 6550  ‘cfv 6554  (class class class)co 7424   ∘_f cof 7688   ↑_m cmap 8855  Fincfn 8974  ℕcn 12264  ℕ₀cn0 12524  Basecbs 17213  ._rcmulr 17267   Σ_g cgsu 17455  Moorecmre 17595  mrClscmrc 17596  ._gcmg 19061  mulGrpcmgp 20117  Ringcrg 20216  CRingccrg 20217   RingHom crh 20451  SubRingcsubrg 20551  AssAlgcasa 21848  AlgSpancasp 21849  algSccascl 21850   mPwSer cmps 21901   mVar cmvr 21902   mPoly cmpl 21903

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 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  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-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7690  df-ofr 7691  df-om 7877  df-1st 8003  df-2nd 8004  df-supp 8175  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-er 8734  df-map 8857  df-pm 8858  df-ixp 8927  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fsupp 9406  df-sup 9485  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12611  df-dec 12730  df-uz 12875  df-fz 13539  df-fzo 13682  df-seq 14022  df-hash 14348  df-struct 17149  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-mulr 17280  df-sca 17282  df-vsca 17283  df-ip 17284  df-tset 17285  df-ple 17286  df-ds 17288  df-hom 17290  df-cco 17291  df-0g 17456  df-gsum 17457  df-prds 17462  df-pws 17464  df-mre 17599  df-mrc 17600  df-acs 17602  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-mhm 18773  df-submnd 18774  df-grp 18931  df-minusg 18932  df-sbg 18933  df-mulg 19062  df-subg 19117  df-ghm 19207  df-cntz 19311  df-cmn 19780  df-abl 19781  df-mgp 20118  df-rng 20136  df-ur 20165  df-srg 20170  df-ring 20218  df-cring 20219  df-rhm 20454  df-subrng 20528  df-subrg 20553  df-lmod 20838  df-lss 20909  df-lsp 20949  df-assa 21851  df-asp 21852  df-ascl 21853  df-psr 21906  df-mvr 21907  df-mpl 21908

This theorem is referenced by:  evlsval2  22102  evlsval3  42031

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