Theorem evlseu 22074

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 2737	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
3		evlseu.c	. . . 4 ⊢ 𝐶 = (Base‘𝑆)
4		eqid 2737	. . . 4 ⊢ {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin} = {𝑧 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑧 “ ℕ) ∈ Fin}
5		eqid 2737	. . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
6		eqid 2737	. . . 4 ⊢ (.g‘(mulGrp‘𝑆)) = (.g‘(mulGrp‘𝑆))
7		eqid 2737	. . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆)
8		evlseu.v	. . . 4 ⊢ 𝑉 = (𝐼 mVar 𝑅)
9		eqid 2737	. . . 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 22073	. . 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 5807	. . . . . . 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 2739	. . . . . 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 5807	. . . . . . 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 2739	. . . . . 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 633	. . . . 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 3565	. . . 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 1115	. . 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 2737	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
26		crngring 20220	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
27	11, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
28	1, 2, 25, 15, 10, 27	mplasclf 22056	. . . . . . . . 9 ⊢ (𝜑 → 𝐴:(Base‘𝑅)⟶(Base‘𝑃))
29	28	ffund 6667	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐴)
30		funcoeqres 6806	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ (𝑚 ∘ 𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹 ∘ ◡𝐴))
31	29, 30	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∘ 𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹 ∘ ◡𝐴))
32	1, 8, 2, 10, 27	mvrf2 21984	. . . . . . . . 9 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑃))
33	32	ffund 6667	. . . . . . . 8 ⊢ (𝜑 → Fun 𝑉)
34		funcoeqres 6806	. . . . . . . 8 ⊢ ((Fun 𝑉 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ◡𝑉))
35	33, 34	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ◡𝑉))
36	31, 35	anim12dan 620	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺)) → ((𝑚 ↾ ran 𝐴) = (𝐹 ∘ ◡𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ◡𝑉)))
37	36	ex 412	. . . . 5 ⊢ (𝜑 → (((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) → ((𝑚 ↾ ran 𝐴) = (𝐹 ∘ ◡𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ◡𝑉))))
38		resundi 5953	. . . . . 6 ⊢ (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉))
39		uneq12 4104	. . . . . 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 2784	. . . . 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 3134	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ (𝑃 RingHom 𝑆)(((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉))))
43		eqtr3 2759	. . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
45	44, 10, 11	psrassa 21964	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg)
46		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
47	44, 8, 46, 10, 27	mvrf 21976	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅)))
48	47	frnd 6671	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
49		eqid 2737	. . . . . . . . . . . . 13 ⊢ (AlgSpan‘(𝐼 mPwSer 𝑅)) = (AlgSpan‘(𝐼 mPwSer 𝑅))
50		eqid 2737	. . . . . . . . . . . . 13 ⊢ (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘(𝐼 mPwSer 𝑅))
51		eqid 2737	. . . . . . . . . . . . 13 ⊢ (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅))) = (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))
52	49, 50, 51, 46	aspval2 21891	. . . . . . . . . . . 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 585	. . . . . . . . . . 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 22033	. . . . . . . . . . 11 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑃))
55	44, 1, 2, 10, 27	mplsubrg 21996	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
56	1, 44, 2	mplval2 21987	. . . . . . . . . . . . . . . 16 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑃))
57	56	subsubrg2 20570	. . . . . . . . . . . . . . 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 6839	. . . . . . . . . . . . 13 ⊢ (𝜑 → (mrCls‘(SubRing‘𝑃)) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))))
60	50, 56	ressascl 21889	. . . . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 = (algSc‘(𝐼 mPwSer 𝑅)))
63	62	rneqd 5888	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐴 = ran (algSc‘(𝐼 mPwSer 𝑅)))
64	63	uneq1d 4108	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran 𝐴 ∪ ran 𝑉) = (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉))
65	59, 64	fveq12d 6842	. . . . . . . . . . . 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 21854	. . . . . . . . . . . . . 14 ⊢ ((𝐼 mPwSer 𝑅) ∈ AssAlg → (𝐼 mPwSer 𝑅) ∈ Ring)
67	46	subrgmre 20568	. . . . . . . . . . . . . 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 6671	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐴 ⊆ (Base‘𝑃))
70	63, 69	eqsstrrd 3958	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran (algSc‘(𝐼 mPwSer 𝑅)) ⊆ (Base‘𝑃))
71	32	frnd 6671	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
72	70, 71	unssd 4133	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉) ⊆ (Base‘𝑃))
73		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
74	51, 73	submrc 17588	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 2773	. . . . . . . . . . 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 2780	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑃) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
78	77	ad2antrr 727	. . . . . . . . 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 22014	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ Ring)
80	2	subrgmre 20568	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ Ring → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
81	79, 80	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
82	81	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
83		simpr 484	. . . . . . . . . 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 20574	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆)) → dom (𝑚 ∩ 𝑛) ∈ (SubRing‘𝑃))
85	84	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → dom (𝑚 ∩ 𝑛) ∈ (SubRing‘𝑃))
86		eqid 2737	. . . . . . . . . . 11 ⊢ (mrCls‘(SubRing‘𝑃)) = (mrCls‘(SubRing‘𝑃))
87	86	mrcsscl 17580	. . . . . . . . . 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 1374	. . . . . . . . 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 3957	. . . . . . . 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 412	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛) → (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
91		simprl 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 ∈ (𝑃 RingHom 𝑆))
92	2, 3	rhmf 20458	. . . . . . . . 9 ⊢ (𝑚 ∈ (𝑃 RingHom 𝑆) → 𝑚:(Base‘𝑃)⟶𝐶)
93		ffn 6663	. . . . . . . . 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 773	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 ∈ (𝑃 RingHom 𝑆))
96	2, 3	rhmf 20458	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑃 RingHom 𝑆) → 𝑛:(Base‘𝑃)⟶𝐶)
97		ffn 6663	. . . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝐴 ⊆ (Base‘𝑃))
100	71	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝑉 ⊆ (Base‘𝑃))
101	99, 100	unssd 4133	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (ran 𝐴 ∪ ran 𝑉) ⊆ (Base‘𝑃))
102		fnreseql 6995	. . . . . . . 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 1374	. . . . . . 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 6994	. . . . . . . 8 ⊢ ((𝑚 Fn (Base‘𝑃) ∧ 𝑛 Fn (Base‘𝑃)) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
105	94, 98, 104	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
106	90, 103, 105	3imtr4d 294	. . . . . 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 3181	. . . 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 5933	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)))
110	109	eqeq1d 2739	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉)) ↔ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉))))
111	110	rmo4 3677	. . . 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 234	. . 3 ⊢ (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)(𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ◡𝐴) ∪ (𝐺 ∘ ◡𝑉)))
113		rmoim 3687	. . 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 3345	. 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 584	1 ⊢ (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ∃!wreu 3341  ∃*wrmo 3342  {crab 3390   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542   ↦ cmpt 5167  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628   ∘ ccom 5629  Fun wfun 6487   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361   ∘_f cof 7623   ↑_m cmap 8767  Fincfn 8887  ℕcn 12168  ℕ₀cn0 12431  Basecbs 17173  ._rcmulr 17215   Σ_g cgsu 17397  Moorecmre 17538  mrClscmrc 17539  ._gcmg 19037  mulGrpcmgp 20115  Ringcrg 20208  CRingccrg 20209   RingHom crh 20443  SubRingcsubrg 20540  AssAlgcasa 21843  AlgSpancasp 21844  algSccascl 21845   mPwSer cmps 21897   mVar cmvr 21898   mPoly cmpl 21899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7625  df-ofr 7626  df-om 7812  df-1st 7936  df-2nd 7937  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-sup 9349  df-oi 9419  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-uz 12783  df-fz 13456  df-fzo 13603  df-seq 13958  df-hash 14287  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-sca 17230  df-vsca 17231  df-ip 17232  df-tset 17233  df-ple 17234  df-ds 17236  df-hom 17238  df-cco 17239  df-0g 17398  df-gsum 17399  df-prds 17404  df-pws 17406  df-mre 17542  df-mrc 17543  df-acs 17545  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-submnd 18746  df-grp 18906  df-minusg 18907  df-sbg 18908  df-mulg 19038  df-subg 19093  df-ghm 19182  df-cntz 19286  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-srg 20162  df-ring 20210  df-cring 20211  df-rhm 20446  df-subrng 20517  df-subrg 20541  df-lmod 20851  df-lss 20921  df-lsp 20961  df-assa 21846  df-asp 21847  df-ascl 21848  df-psr 21902  df-mvr 21903  df-mpl 21904

This theorem is referenced by:  evlsval2  22078  evlsval3  22080