evlseu - Metamath Proof Explorer

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Theorem evlseu 21865

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 2730	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
3		evlseu.c	. . . 4 ⊢ 𝐶 = (Base‘𝑆)
4		eqid 2730	. . . 4 ⊢ {𝑧 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin} = {𝑧 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑧 “ ℕ) ∈ Fin}
5		eqid 2730	. . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
6		eqid 2730	. . . 4 ⊢ (._g‘(mulGrp‘𝑆)) = (._g‘(mulGrp‘𝑆))
7		eqid 2730	. . . 4 ⊢ (._r‘𝑆) = (._r‘𝑆)
8		evlseu.v	. . . 4 ⊢ 𝑉 = (𝐼 mVar 𝑅)
9		eqid 2730	. . . 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))))))
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 21864	. . 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))
17		coeq1 5856	. . . . . . 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))
18	17	eqeq1d 2732	. . . . . 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))
19		coeq1 5856	. . . . . . 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))
20	19	eqeq1d 2732	. . . . . 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))
21	18, 20	anbi12d 629	. . . . 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)))
22	21	rspcev 3611	. . . 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))
23	22	3impb 1113	. . 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))
24	16, 23	syl 17	. 2 ⊢ (𝜑 → ∃𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))
25		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
26		crngring 20139	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
27	11, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
28	1, 2, 25, 15, 10, 27	mplasclf 21845	. . . . . . . . 9 ⊢ (𝜑 → 𝐴:(Base‘𝑅)⟶(Base‘𝑃))
29	28	ffund 6720	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐴)
30		funcoeqres 6863	. . . . . . . 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 21771	. . . . . . . . 9 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑃))
33	32	ffund 6720	. . . . . . . 8 ⊢ (𝜑 → Fun 𝑉)
34		funcoeqres 6863	. . . . . . . 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 5994	. . . . . 6 ⊢ (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉))
39		uneq12 4157	. . . . . 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 2782	. . . . 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 3148	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ (𝑃 RingHom 𝑆)(((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ^◡𝐴) ∪ (𝐺 ∘ ^◡𝑉))))
43		eqtr3 2756	. . . . . 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 2730	. . . . . . . . . . . . 13 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
45	44, 10, 11	psrassa 21753	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg)
46		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
47	44, 8, 46, 10, 27	mvrf 21763	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅)))
48	47	frnd 6724	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
49		eqid 2730	. . . . . . . . . . . . 13 ⊢ (AlgSpan‘(𝐼 mPwSer 𝑅)) = (AlgSpan‘(𝐼 mPwSer 𝑅))
50		eqid 2730	. . . . . . . . . . . . 13 ⊢ (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘(𝐼 mPwSer 𝑅))
51		eqid 2730	. . . . . . . . . . . . 13 ⊢ (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅))) = (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))
52	49, 50, 51, 46	aspval2 21671	. . . . . . . . . . . 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 21816	. . . . . . . . . . 11 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑃))
55	44, 1, 2, 10, 27	mplsubrg 21783	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
56	1, 44, 2	mplval2 21774	. . . . . . . . . . . . . . . 16 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾_s (Base‘𝑃))
57	56	subsubrg2 20489	. . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . 13 ⊢ (𝜑 → (mrCls‘(SubRing‘𝑃)) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))))
60	50, 56	ressascl 21669	. . . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 = (algSc‘(𝐼 mPwSer 𝑅)))
63	62	rneqd 5936	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐴 = ran (algSc‘(𝐼 mPwSer 𝑅)))
64	63	uneq1d 4161	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran 𝐴 ∪ ran 𝑉) = (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉))
65	59, 64	fveq12d 6897	. . . . . . . . . . . 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 21635	. . . . . . . . . . . . . 14 ⊢ ((𝐼 mPwSer 𝑅) ∈ AssAlg → (𝐼 mPwSer 𝑅) ∈ Ring)
67	46	subrgmre 20487	. . . . . . . . . . . . . 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 6724	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐴 ⊆ (Base‘𝑃))
70	63, 69	eqsstrrd 4020	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran (algSc‘(𝐼 mPwSer 𝑅)) ⊆ (Base‘𝑃))
71	32	frnd 6724	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
72	70, 71	unssd 4185	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉) ⊆ (Base‘𝑃))
73		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
74	51, 73	submrc 17576	. . . . . . . . . . . . 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 1369	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 2778	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑃) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
78	77	ad2antrr 722	. . . . . . . . 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 21797	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
80	10, 27, 79	syl2anc 582	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ Ring)
81	2	subrgmre 20487	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ Ring → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
82	80, 81	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
83	82	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
84		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))
85		rhmeql 20493	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆)) → dom (𝑚 ∩ 𝑛) ∈ (SubRing‘𝑃))
86	85	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛)) → dom (𝑚 ∩ 𝑛) ∈ (SubRing‘𝑃))
87		eqid 2730	. . . . . . . . . . 11 ⊢ (mrCls‘(SubRing‘𝑃)) = (mrCls‘(SubRing‘𝑃))
88	87	mrcsscl 17568	. . . . . . . . . 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 1369	. . . . . . . . 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 4019	. . . . . . . 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 411	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚 ∩ 𝑛) → (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
92		simprl 767	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 ∈ (𝑃 RingHom 𝑆))
93	2, 3	rhmf 20376	. . . . . . . . 9 ⊢ (𝑚 ∈ (𝑃 RingHom 𝑆) → 𝑚:(Base‘𝑃)⟶𝐶)
94		ffn 6716	. . . . . . . . 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 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 ∈ (𝑃 RingHom 𝑆))
97	2, 3	rhmf 20376	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑃 RingHom 𝑆) → 𝑛:(Base‘𝑃)⟶𝐶)
98		ffn 6716	. . . . . . . . 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 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝐴 ⊆ (Base‘𝑃))
101	71	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝑉 ⊆ (Base‘𝑃))
102	100, 101	unssd 4185	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (ran 𝐴 ∪ ran 𝑉) ⊆ (Base‘𝑃))
103		fnreseql 7048	. . . . . . . 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 1369	. . . . . . 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 7047	. . . . . . . 8 ⊢ ((𝑚 Fn (Base‘𝑃) ∧ 𝑛 Fn (Base‘𝑃)) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
106	95, 99, 105	syl2anc 582	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚 ∩ 𝑛)))
107	91, 104, 106	3imtr4d 293	. . . . . 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 3198	. . . 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 5974	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)))
111	110	eqeq1d 2732	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ^◡𝐴) ∪ (𝐺 ∘ ^◡𝑉)) ↔ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹 ∘ ^◡𝐴) ∪ (𝐺 ∘ ^◡𝑉))))
112	111	rmo4 3725	. . . 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 3735	. . 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 3376	. 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 581	1 ⊢ (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 ∃!wreu 3372 ∃*wrmo 3373 {crab 3430 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4601 ↦ cmpt 5230 ^◡ccnv 5674 dom cdm 5675 ran crn 5676 ↾ cres 5677 “ cima 5678 ∘ ccom 5679 Fun wfun 6536 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ∘_f cof 7670 ↑_m cmap 8822 Fincfn 8941 ℕcn 12216 ℕ₀cn0 12476 Basecbs 17148 ._rcmulr 17202 Σ_g cgsu 17390 Moorecmre 17530 mrClscmrc 17531 ._gcmg 18986 mulGrpcmgp 20028 Ringcrg 20127 CRingccrg 20128 RingHom crh 20360 SubRingcsubrg 20457 AssAlgcasa 21624 AlgSpancasp 21625 algSccascl 21626 mPwSer cmps 21676 mVar cmvr 21677 mPoly cmpl 21678

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-hom 17225 df-cco 17226 df-0g 17391 df-gsum 17392 df-prds 17397 df-pws 17399 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-subg 19039 df-ghm 19128 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-srg 20081 df-ring 20129 df-cring 20130 df-rhm 20363 df-subrng 20434 df-subrg 20459 df-lmod 20616 df-lss 20687 df-lsp 20727 df-assa 21627 df-asp 21628 df-ascl 21629 df-psr 21681 df-mvr 21682 df-mpl 21683

This theorem is referenced by: evlsval2 21869 evlsval3 41433

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