Theorem evlseu 21293

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 2738	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
3		evlseu.c	. . . 4 ⊢ 𝐶 = (Base‘𝑆)
4		eqid 2738	. . . 4 ⊢ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} = {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin}
5		eqid 2738	. . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
6		eqid 2738	. . . 4 ⊢ (.g‘(mulGrp‘𝑆)) = (.g‘(mulGrp‘𝑆))
7		eqid 2738	. . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆)
8		evlseu.v	. . . 4 ⊢ 𝑉 = (𝐼 mVar 𝑅)
9		eqid 2738	. . . 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 21292	. . 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 5766	. . . . . . 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 2740	. . . . . 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 5766	. . . . . . 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 2740	. . . . . 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 631	. . . . 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 3561	. . . 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 1114	. . 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 2738	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
26		crngring 19795	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
27	11, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
28	1, 2, 25, 15, 10, 27	mplasclf 21273	. . . . . . . . 9 ⊢ (𝜑 → 𝐴:(Base‘𝑅)⟶(Base‘𝑃))
29	28	ffund 6604	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐴)
30		funcoeqres 6747	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ (𝑚 ∘ 𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹 ∘ ◡𝐴))
31	29, 30	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∘ 𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹 ∘ ◡𝐴))
32	1, 8, 2, 10, 27	mvrf2 21268	. . . . . . . . 9 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑃))
33	32	ffund 6604	. . . . . . . 8 ⊢ (𝜑 → Fun 𝑉)
34		funcoeqres 6747	. . . . . . . 8 ⊢ ((Fun 𝑉 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ◡𝑉))
35	33, 34	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ◡𝑉))
36	31, 35	anim12dan 619	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺)) → ((𝑚 ↾ ran 𝐴) = (𝐹 ∘ ◡𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ◡𝑉)))
37	36	ex 413	. . . . 5 ⊢ (𝜑 → (((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) → ((𝑚 ↾ ran 𝐴) = (𝐹 ∘ ◡𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ◡𝑉))))
38		resundi 5905	. . . . . 6 ⊢ (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉))
39		uneq12 4092	. . . . . 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 2790	. . . . 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 3104	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ (𝑃 RingHom 𝑆)(((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉))))
43		eqtr3 2764	. . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
45	44, 10, 11	psrassa 21183	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg)
46		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
47	44, 8, 46, 10, 27	mvrf 21193	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅)))
48	47	frnd 6608	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
49		eqid 2738	. . . . . . . . . . . . 13 ⊢ (AlgSpan‘(𝐼 mPwSer 𝑅)) = (AlgSpan‘(𝐼 mPwSer 𝑅))
50		eqid 2738	. . . . . . . . . . . . 13 ⊢ (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘(𝐼 mPwSer 𝑅))
51		eqid 2738	. . . . . . . . . . . . 13 ⊢ (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅))) = (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))
52	49, 50, 51, 46	aspval2 21102	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 21243	. . . . . . . . . . 11 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑃))
55	44, 1, 2, 10, 27	mplsubrg 21211	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
56	1, 44, 2	mplval2 21202	. . . . . . . . . . . . . . . 16 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑃))
57	56	subsubrg2 20051	. . . . . . . . . . . . . . 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 6778	. . . . . . . . . . . . 13 ⊢ (𝜑 → (mrCls‘(SubRing‘𝑃)) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))))
60	50, 56	ressascl 21100	. . . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 = (algSc‘(𝐼 mPwSer 𝑅)))
63	62	rneqd 5847	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐴 = ran (algSc‘(𝐼 mPwSer 𝑅)))
64	63	uneq1d 4096	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran 𝐴 ∪ ran 𝑉) = (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉))
65	59, 64	fveq12d 6781	. . . . . . . . . . . 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 21068	. . . . . . . . . . . . . 14 ⊢ ((𝐼 mPwSer 𝑅) ∈ AssAlg → (𝐼 mPwSer 𝑅) ∈ Ring)
67	46	subrgmre 20048	. . . . . . . . . . . . . 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 6608	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐴 ⊆ (Base‘𝑃))
70	63, 69	eqsstrrd 3960	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran (algSc‘(𝐼 mPwSer 𝑅)) ⊆ (Base‘𝑃))
71	32	frnd 6608	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
72	70, 71	unssd 4120	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉) ⊆ (Base‘𝑃))
73		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
74	51, 73	submrc 17337	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 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 2779	. . . . . . . . . . 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 2786	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑃) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
78	77	ad2antrr 723	. . . . . . . . 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	mplring 21224	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
80	10, 27, 79	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ Ring)
81	2	subrgmre 20048	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ Ring → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
82	80, 81	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
83	82	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
84		simpr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛))
85		rhmeql 20054	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆)) → dom (𝑚 ∩ 𝑛) ∈ (SubRing‘𝑃))
86	85	ad2antlr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → dom (𝑚 ∩ 𝑛) ∈ (SubRing‘𝑃))
87		eqid 2738	. . . . . . . . . . 11 ⊢ (mrCls‘(SubRing‘𝑃)) = (mrCls‘(SubRing‘𝑃))
88	87	mrcsscl 17329	. . . . . . . . . 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))
89	83, 84, 86, 88	syl3anc 1370	. . . . . . . . 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))
90	78, 89	eqsstrd 3959	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛))
91	90	ex 413	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛) → (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
92		simprl 768	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 ∈ (𝑃 RingHom 𝑆))
93	2, 3	rhmf 19970	. . . . . . . . 9 ⊢ (𝑚 ∈ (𝑃 RingHom 𝑆) → 𝑚:(Base‘𝑃)⟶𝐶)
94		ffn 6600	. . . . . . . . 9 ⊢ (𝑚:(Base‘𝑃)⟶𝐶 → 𝑚 Fn (Base‘𝑃))
95	92, 93, 94	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 Fn (Base‘𝑃))
96		simprr 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 ∈ (𝑃 RingHom 𝑆))
97	2, 3	rhmf 19970	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑃 RingHom 𝑆) → 𝑛:(Base‘𝑃)⟶𝐶)
98		ffn 6600	. . . . . . . . 9 ⊢ (𝑛:(Base‘𝑃)⟶𝐶 → 𝑛 Fn (Base‘𝑃))
99	96, 97, 98	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 Fn (Base‘𝑃))
100	69	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝐴 ⊆ (Base‘𝑃))
101	71	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝑉 ⊆ (Base‘𝑃))
102	100, 101	unssd 4120	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (ran 𝐴 ∪ ran 𝑉) ⊆ (Base‘𝑃))
103		fnreseql 6925	. . . . . . . 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)))
104	95, 99, 102, 103	syl3anc 1370	. . . . . . 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)))
105		fneqeql2 6924	. . . . . . . 8 ⊢ ((𝑚 Fn (Base‘𝑃) ∧ 𝑛 Fn (Base‘𝑃)) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
106	95, 99, 105	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
107	91, 104, 106	3imtr4d 294	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) → 𝑚 = 𝑛))
108	43, 107	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))
109	108	ralrimivva 3123	. . . 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))
110		reseq1 5885	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)))
111	110	eqeq1d 2740	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉)) ↔ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉))))
112	111	rmo4 3665	. . . 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))
113	109, 112	sylibr 233	. . 3 ⊢ (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)(𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉)))
114		rmoim 3675	. . 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)))
115	42, 113, 114	sylc 65	. 2 ⊢ (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))
116		reu5 3361	. 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)))
117	24, 115, 116	sylanbrc 583	1 ⊢ (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  ∃!wreu 3066  ∃*wrmo 3067  {crab 3068   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533   ↦ cmpt 5157  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   ↾ cres 5591   “ cima 5592   ∘ ccom 5593  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∘_f cof 7531   ↑_m cmap 8615  Fincfn 8733  ℕcn 11973  ℕ₀cn0 12233  Basecbs 16912  ._rcmulr 16963   Σ_g cgsu 17151  Moorecmre 17291  mrClscmrc 17292  ._gcmg 18700  mulGrpcmgp 19720  Ringcrg 19783  CRingccrg 19784   RingHom crh 19956  SubRingcsubrg 20020  AssAlgcasa 21057  AlgSpancasp 21058  algSccascl 21059   mPwSer cmps 21107   mVar cmvr 21108   mPoly cmpl 21109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-ofr 7534  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-seq 13722  df-hash 14045  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-tset 16981  df-0g 17152  df-gsum 17153  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-mulg 18701  df-subg 18752  df-ghm 18832  df-cntz 18923  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-srg 19742  df-ring 19785  df-cring 19786  df-rnghom 19959  df-subrg 20022  df-lmod 20125  df-lss 20194  df-lsp 20234  df-assa 21060  df-asp 21061  df-ascl 21062  df-psr 21112  df-mvr 21113  df-mpl 21114

This theorem is referenced by:  evlsval2  21297  evlsval3  40272