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Theorem evlseu 19516

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.)

**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 ∈ V)
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)

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 2622	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
3		evlseu.c	. . . 4 ⊢ 𝐶 = (Base‘𝑆)
4		eqid 2622	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2622	. . . 4 ⊢ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} = {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin}
6		eqid 2622	. . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
7		eqid 2622	. . . 4 ⊢ (._g‘(mulGrp‘𝑆)) = (._g‘(mulGrp‘𝑆))
8		eqid 2622	. . . 4 ⊢ (._r‘𝑆) = (._r‘𝑆)
9		evlseu.v	. . . 4 ⊢ 𝑉 = (𝐼 mVar 𝑅)
10		eqid 2622	. . . 4 ⊢ (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺))))))
11		evlseu.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
12		evlseu.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing)
13		evlseu.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ CRing)
14		evlseu.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))
15		evlseu.g	. . . 4 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶)
16		evlseu.a	. . . 4 ⊢ 𝐴 = (algSc‘𝑃)
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	evlslem1 19515	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∈ (𝑃 RingHom 𝑆) ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺))
18		coeq1 5279	. . . . . . 7 ⊢ (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) → (𝑚 ∘ 𝐴) = ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴))
19	18	eqeq1d 2624	. . . . . 6 ⊢ (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) → ((𝑚 ∘ 𝐴) = 𝐹 ↔ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹))
20		coeq1 5279	. . . . . . 7 ⊢ (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) → (𝑚 ∘ 𝑉) = ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉))
21	20	eqeq1d 2624	. . . . . 6 ⊢ (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) → ((𝑚 ∘ 𝑉) = 𝐺 ↔ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺))
22	19, 21	anbi12d 747	. . . . 5 ⊢ (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) → (((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) ↔ (((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺)))
23	22	rspcev 3309	. . . 4 ⊢ (((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∈ (𝑃 RingHom 𝑆) ∧ (((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺)) → ∃𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))
24	23	3impb 1260	. . 3 ⊢ (((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∈ (𝑃 RingHom 𝑆) ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σ_g (𝑦 ∈ {𝑧 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥‘𝑦))(._r‘𝑆)((mulGrp‘𝑆) Σ_g (𝑦 ∘_𝑓 (._g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺) → ∃𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))
25	17, 24	syl 17	. 2 ⊢ (𝜑 → ∃𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))
26		crngring 18558	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
27	12, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
28		eqid 2622	. . . . . . . . . . 11 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
29	1	mplring 19452	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
30	1	mpllmod 19451	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ Ring) → 𝑃 ∈ LMod)
31		eqid 2622	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
32	16, 28, 29, 30, 31, 2	asclf 19337	. . . . . . . . . 10 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ Ring) → 𝐴:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
33	11, 27, 32	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → 𝐴:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
34		ffun 6048	. . . . . . . . 9 ⊢ (𝐴:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃) → Fun 𝐴)
35	33, 34	syl 17	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐴)
36		funcoeqres 6167	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ (𝑚 ∘ 𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹 ∘ ^◡𝐴))
37	35, 36	sylan 488	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∘ 𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹 ∘ ^◡𝐴))
38	1, 9, 2, 11, 27	mvrf2 19492	. . . . . . . . 9 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑃))
39		ffun 6048	. . . . . . . . 9 ⊢ (𝑉:𝐼⟶(Base‘𝑃) → Fun 𝑉)
40	38, 39	syl 17	. . . . . . . 8 ⊢ (𝜑 → Fun 𝑉)
41		funcoeqres 6167	. . . . . . . 8 ⊢ ((Fun 𝑉 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ^◡𝑉))
42	40, 41	sylan 488	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ^◡𝑉))
43	37, 42	anim12dan 882	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺)) → ((𝑚 ↾ ran 𝐴) = (𝐹 ∘ ^◡𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ^◡𝑉)))
44	43	ex 450	. . . . 5 ⊢ (𝜑 → (((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) → ((𝑚 ↾ ran 𝐴) = (𝐹 ∘ ^◡𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ^◡𝑉))))
45		resundi 5410	. . . . . 6 ⊢ (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉))
46		uneq12 3762	. . . . . 6 ⊢ (((𝑚 ↾ ran 𝐴) = (𝐹 ∘ ^◡𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ^◡𝑉)) → ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉)) = ((𝐹 ∘ ^◡𝐴) ∪ (𝐺 ∘ ^◡𝑉)))
47	45, 46	syl5eq 2668	. . . . 5 ⊢ (((𝑚 ↾ ran 𝐴) = (𝐹 ∘ ^◡𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺 ∘ ^◡𝑉)) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ^◡𝐴) ∪ (𝐺 ∘ ^◡𝑉)))
48	44, 47	syl6 35	. . . 4 ⊢ (𝜑 → (((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ^◡𝐴) ∪ (𝐺 ∘ ^◡𝑉))))
49	48	ralrimivw 2967	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ (𝑃 RingHom 𝑆)(((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ^◡𝐴) ∪ (𝐺 ∘ ^◡𝑉))))
50		eqtr3 2643	. . . . . 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)))
51		eqid 2622	. . . . . . . . . . . . 13 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
52	51, 11, 12	psrassa 19414	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg)
53		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
54	51, 9, 53, 11, 27	mvrf 19424	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅)))
55		frn 6053	. . . . . . . . . . . . 13 ⊢ (𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅)) → ran 𝑉 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
56	54, 55	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
57		eqid 2622	. . . . . . . . . . . . 13 ⊢ (AlgSpan‘(𝐼 mPwSer 𝑅)) = (AlgSpan‘(𝐼 mPwSer 𝑅))
58		eqid 2622	. . . . . . . . . . . . 13 ⊢ (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘(𝐼 mPwSer 𝑅))
59		eqid 2622	. . . . . . . . . . . . 13 ⊢ (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅))) = (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))
60	57, 58, 59, 53	aspval2 19347	. . . . . . . . . . . 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)))
61	52, 56, 60	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)))
62	1, 51, 9, 57, 11, 12	mplbas2 19470	. . . . . . . . . . 11 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑃))
63	51, 1, 2, 11, 27	mplsubrg 19440	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
64	1, 51, 2	mplval2 19431	. . . . . . . . . . . . . . . 16 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾_s (Base‘𝑃))
65	64	subsubrg2 18807	. . . . . . . . . . . . . . 15 ⊢ ((Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → (SubRing‘𝑃) = ((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
66	63, 65	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (SubRing‘𝑃) = ((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
67	66	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝜑 → (mrCls‘(SubRing‘𝑃)) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))))
68	58, 64	ressascl 19344	. . . . . . . . . . . . . . . . 17 ⊢ ((Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘𝑃))
69	63, 68	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘𝑃))
70	69, 16	syl6reqr 2675	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 = (algSc‘(𝐼 mPwSer 𝑅)))
71	70	rneqd 5353	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐴 = ran (algSc‘(𝐼 mPwSer 𝑅)))
72	71	uneq1d 3766	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran 𝐴 ∪ ran 𝑉) = (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉))
73	67, 72	fveq12d 6197	. . . . . . . . . . . 12 ⊢ (𝜑 → ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)) = ((mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)))
74		assaring 19320	. . . . . . . . . . . . . 14 ⊢ ((𝐼 mPwSer 𝑅) ∈ AssAlg → (𝐼 mPwSer 𝑅) ∈ Ring)
75	53	subrgmre 18804	. . . . . . . . . . . . . 14 ⊢ ((𝐼 mPwSer 𝑅) ∈ Ring → (SubRing‘(𝐼 mPwSer 𝑅)) ∈ (Moore‘(Base‘(𝐼 mPwSer 𝑅))))
76	52, 74, 75	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (SubRing‘(𝐼 mPwSer 𝑅)) ∈ (Moore‘(Base‘(𝐼 mPwSer 𝑅))))
77		frn 6053	. . . . . . . . . . . . . . . 16 ⊢ (𝐴:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃) → ran 𝐴 ⊆ (Base‘𝑃))
78	33, 77	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐴 ⊆ (Base‘𝑃))
79	71, 78	eqsstr3d 3640	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran (algSc‘(𝐼 mPwSer 𝑅)) ⊆ (Base‘𝑃))
80		frn 6053	. . . . . . . . . . . . . . 15 ⊢ (𝑉:𝐼⟶(Base‘𝑃) → ran 𝑉 ⊆ (Base‘𝑃))
81	38, 80	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
82	79, 81	unssd 3789	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉) ⊆ (Base‘𝑃))
83		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
84	59, 83	submrc 16288	. . . . . . . . . . . . 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)))
85	76, 63, 82, 84	syl3anc 1326	. . . . . . . . . . . 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)))
86	73, 85	eqtr2d 2657	. . . . . . . . . . 11 ⊢ (𝜑 → ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
87	61, 62, 86	3eqtr3d 2664	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑃) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
88	87	ad2antrr 762	. . . . . . . . 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)))
89	11, 27, 29	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ Ring)
90	2	subrgmre 18804	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ Ring → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
91	89, 90	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
92	91	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
93		simpr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛))
94		rhmeql 18810	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆)) → dom (𝑚 ∩ 𝑛) ∈ (SubRing‘𝑃))
95	94	ad2antlr 763	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → dom (𝑚 ∩ 𝑛) ∈ (SubRing‘𝑃))
96		eqid 2622	. . . . . . . . . . 11 ⊢ (mrCls‘(SubRing‘𝑃)) = (mrCls‘(SubRing‘𝑃))
97	96	mrcsscl 16280	. . . . . . . . . 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))
98	92, 93, 95, 97	syl3anc 1326	. . . . . . . . 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))
99	88, 98	eqsstrd 3639	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛))
100	99	ex 450	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛) → (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
101		simprl 794	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 ∈ (𝑃 RingHom 𝑆))
102	2, 3	rhmf 18726	. . . . . . . . 9 ⊢ (𝑚 ∈ (𝑃 RingHom 𝑆) → 𝑚:(Base‘𝑃)⟶𝐶)
103		ffn 6045	. . . . . . . . 9 ⊢ (𝑚:(Base‘𝑃)⟶𝐶 → 𝑚 Fn (Base‘𝑃))
104	101, 102, 103	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 Fn (Base‘𝑃))
105		simprr 796	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 ∈ (𝑃 RingHom 𝑆))
106	2, 3	rhmf 18726	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑃 RingHom 𝑆) → 𝑛:(Base‘𝑃)⟶𝐶)
107		ffn 6045	. . . . . . . . 9 ⊢ (𝑛:(Base‘𝑃)⟶𝐶 → 𝑛 Fn (Base‘𝑃))
108	105, 106, 107	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 Fn (Base‘𝑃))
109	78	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝐴 ⊆ (Base‘𝑃))
110	81	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝑉 ⊆ (Base‘𝑃))
111	109, 110	unssd 3789	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (ran 𝐴 ∪ ran 𝑉) ⊆ (Base‘𝑃))
112		fnreseql 6327	. . . . . . . 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)))
113	104, 108, 111, 112	syl3anc 1326	. . . . . . 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)))
114		fneqeql2 6326	. . . . . . . 8 ⊢ ((𝑚 Fn (Base‘𝑃) ∧ 𝑛 Fn (Base‘𝑃)) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
115	104, 108, 114	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
116	100, 113, 115	3imtr4d 283	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) → 𝑚 = 𝑛))
117	50, 116	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))
118	117	ralrimivva 2971	. . . 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))
119		reseq1 5390	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)))
120	119	eqeq1d 2624	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ^◡𝐴) ∪ (𝐺 ∘ ^◡𝑉)) ↔ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ^◡𝐴) ∪ (𝐺 ∘ ^◡𝑉))))
121	120	rmo4 3399	. . . 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))
122	118, 121	sylibr 224	. . 3 ⊢ (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)(𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ^◡𝐴) ∪ (𝐺 ∘ ^◡𝑉)))
123		rmoim 3407	. . 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)))
124	49, 122, 123	sylc 65	. 2 ⊢ (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))
125		reu5 3159	. 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)))
126	25, 124, 125	sylanbrc 698	1 ⊢ (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ∃!wreu 2914 ∃*wrmo 2915 {crab 2916 Vcvv 3200 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ↦ cmpt 4729 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 ∘ ccom 5118 Fun wfun 5882 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘_𝑓 cof 6895 ↑_𝑚 cmap 7857 Fincfn 7955 ℕcn 11020 ℕ₀cn0 11292 Basecbs 15857 ._rcmulr 15942 Scalarcsca 15944 Σ_g cgsu 16101 Moorecmre 16242 mrClscmrc 16243 ._gcmg 17540 mulGrpcmgp 18489 Ringcrg 18547 CRingccrg 18548 RingHom crh 18712 SubRingcsubrg 18776 AssAlgcasa 19309 AlgSpancasp 19310 algSccascl 19311 mPwSer cmps 19351 mVar cmvr 19352 mPoly cmpl 19353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-tset 15960 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-srg 18506 df-ring 18549 df-cring 18550 df-rnghom 18715 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-assa 19312 df-asp 19313 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358

This theorem is referenced by: evlsval2 19520

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