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Theorem evlseu 22107
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 (𝜑𝐼𝑊)
evlseu.r (𝜑𝑅 ∈ CRing)
evlseu.s (𝜑𝑆 ∈ CRing)
evlseu.f (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))
evlseu.g (𝜑𝐺:𝐼𝐶)
Assertion
Ref Expression
evlseu (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 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.
StepHypRef 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 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} = {𝑧 ∈ (ℕ0m 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 (.r𝑆) = (.r𝑆)
8 evlseu.v . . . 4 𝑉 = (𝐼 mVar 𝑅)
9 eqid 2737 . . . 4 (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺))))))
10 evlseu.i . . . 4 (𝜑𝐼𝑊)
11 evlseu.r . . . 4 (𝜑𝑅 ∈ CRing)
12 evlseu.s . . . 4 (𝜑𝑆 ∈ CRing)
13 evlseu.f . . . 4 (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))
14 evlseu.g . . . 4 (𝜑𝐺:𝐼𝐶)
15 evlseu.a . . . 4 𝐴 = (algSc‘𝑃)
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15evlslem1 22106 . . 3 (𝜑 → ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∈ (𝑃 RingHom 𝑆) ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺))
17 coeq1 5868 . . . . . . 7 (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) → (𝑚𝐴) = ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴))
1817eqeq1d 2739 . . . . . 6 (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) → ((𝑚𝐴) = 𝐹 ↔ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹))
19 coeq1 5868 . . . . . . 7 (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) → (𝑚𝑉) = ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉))
2019eqeq1d 2739 . . . . . 6 (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) → ((𝑚𝑉) = 𝐺 ↔ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺))
2118, 20anbi12d 632 . . . . 5 (𝑚 = (𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) → (((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) ↔ (((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺)))
2221rspcev 3622 . . . 4 (((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∈ (𝑃 RingHom 𝑆) ∧ (((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺)) → ∃𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
23223impb 1115 . . 3 (((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∈ (𝑃 RingHom 𝑆) ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((𝑥 ∈ (Base‘𝑃) ↦ (𝑆 Σg (𝑦 ∈ {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} ↦ ((𝐹‘(𝑥𝑦))(.r𝑆)((mulGrp‘𝑆) Σg (𝑦f (.g‘(mulGrp‘𝑆))𝐺)))))) ∘ 𝑉) = 𝐺) → ∃𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
2416, 23syl 17 . 2 (𝜑 → ∃𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
25 eqid 2737 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
26 crngring 20242 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2711, 26syl 17 . . . . . . . . . 10 (𝜑𝑅 ∈ Ring)
281, 2, 25, 15, 10, 27mplasclf 22089 . . . . . . . . 9 (𝜑𝐴:(Base‘𝑅)⟶(Base‘𝑃))
2928ffund 6740 . . . . . . . 8 (𝜑 → Fun 𝐴)
30 funcoeqres 6879 . . . . . . . 8 ((Fun 𝐴 ∧ (𝑚𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹𝐴))
3129, 30sylan 580 . . . . . . 7 ((𝜑 ∧ (𝑚𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹𝐴))
321, 8, 2, 10, 27mvrf2 22013 . . . . . . . . 9 (𝜑𝑉:𝐼⟶(Base‘𝑃))
3332ffund 6740 . . . . . . . 8 (𝜑 → Fun 𝑉)
34 funcoeqres 6879 . . . . . . . 8 ((Fun 𝑉 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺𝑉))
3533, 34sylan 580 . . . . . . 7 ((𝜑 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺𝑉))
3631, 35anim12dan 619 . . . . . 6 ((𝜑 ∧ ((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺)) → ((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉)))
3736ex 412 . . . . 5 (𝜑 → (((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) → ((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉))))
38 resundi 6011 . . . . . 6 (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉))
39 uneq12 4163 . . . . . 6 (((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉)) → ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)))
4038, 39eqtrid 2789 . . . . 5 (((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉)) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)))
4137, 40syl6 35 . . . 4 (𝜑 → (((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))))
4241ralrimivw 3150 . . 3 (𝜑 → ∀𝑚 ∈ (𝑃 RingHom 𝑆)(((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))))
43 eqtr3 2763 . . . . . 6 (((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)) ∧ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)))
44 eqid 2737 . . . . . . . . . . . . 13 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
4544, 10, 11psrassa 21993 . . . . . . . . . . . 12 (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg)
46 eqid 2737 . . . . . . . . . . . . . 14 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
4744, 8, 46, 10, 27mvrf 22005 . . . . . . . . . . . . 13 (𝜑𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅)))
4847frnd 6744 . . . . . . . . . . . 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)))
5249, 50, 51, 46aspval2 21918 . . . . . . . . . . . 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)))
5345, 48, 52syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)))
541, 44, 8, 49, 10, 11mplbas2 22060 . . . . . . . . . . 11 (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑃))
5544, 1, 2, 10, 27mplsubrg 22025 . . . . . . . . . . . . . . 15 (𝜑 → (Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
561, 44, 2mplval2 22016 . . . . . . . . . . . . . . . 16 𝑃 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑃))
5756subsubrg2 20599 . . . . . . . . . . . . . . 15 ((Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → (SubRing‘𝑃) = ((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
5855, 57syl 17 . . . . . . . . . . . . . 14 (𝜑 → (SubRing‘𝑃) = ((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
5958fveq2d 6910 . . . . . . . . . . . . 13 (𝜑 → (mrCls‘(SubRing‘𝑃)) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))))
6050, 56ressascl 21916 . . . . . . . . . . . . . . . . 17 ((Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘𝑃))
6155, 60syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘𝑃))
6215, 61eqtr4id 2796 . . . . . . . . . . . . . . 15 (𝜑𝐴 = (algSc‘(𝐼 mPwSer 𝑅)))
6362rneqd 5949 . . . . . . . . . . . . . 14 (𝜑 → ran 𝐴 = ran (algSc‘(𝐼 mPwSer 𝑅)))
6463uneq1d 4167 . . . . . . . . . . . . 13 (𝜑 → (ran 𝐴 ∪ ran 𝑉) = (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉))
6559, 64fveq12d 6913 . . . . . . . . . . . 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 21881 . . . . . . . . . . . . . 14 ((𝐼 mPwSer 𝑅) ∈ AssAlg → (𝐼 mPwSer 𝑅) ∈ Ring)
6746subrgmre 20597 . . . . . . . . . . . . . 14 ((𝐼 mPwSer 𝑅) ∈ Ring → (SubRing‘(𝐼 mPwSer 𝑅)) ∈ (Moore‘(Base‘(𝐼 mPwSer 𝑅))))
6845, 66, 673syl 18 . . . . . . . . . . . . 13 (𝜑 → (SubRing‘(𝐼 mPwSer 𝑅)) ∈ (Moore‘(Base‘(𝐼 mPwSer 𝑅))))
6928frnd 6744 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐴 ⊆ (Base‘𝑃))
7063, 69eqsstrrd 4019 . . . . . . . . . . . . . 14 (𝜑 → ran (algSc‘(𝐼 mPwSer 𝑅)) ⊆ (Base‘𝑃))
7132frnd 6744 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
7270, 71unssd 4192 . . . . . . . . . . . . 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)))
7451, 73submrc 17671 . . . . . . . . . . . . 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)))
7568, 55, 72, 74syl3anc 1373 . . . . . . . . . . . 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)))
7665, 75eqtr2d 2778 . . . . . . . . . . 11 (𝜑 → ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
7753, 54, 763eqtr3d 2785 . . . . . . . . . 10 (𝜑 → (Base‘𝑃) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
7877ad2antrr 726 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (Base‘𝑃) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
791, 10, 27mplringd 22043 . . . . . . . . . . . 12 (𝜑𝑃 ∈ Ring)
802subrgmre 20597 . . . . . . . . . . . 12 (𝑃 ∈ Ring → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
8179, 80syl 17 . . . . . . . . . . 11 (𝜑 → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
8281ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
83 simpr 484 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛))
84 rhmeql 20603 . . . . . . . . . . 11 ((𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆)) → dom (𝑚𝑛) ∈ (SubRing‘𝑃))
8584ad2antlr 727 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → dom (𝑚𝑛) ∈ (SubRing‘𝑃))
86 eqid 2737 . . . . . . . . . . 11 (mrCls‘(SubRing‘𝑃)) = (mrCls‘(SubRing‘𝑃))
8786mrcsscl 17663 . . . . . . . . . 10 (((SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛) ∧ dom (𝑚𝑛) ∈ (SubRing‘𝑃)) → ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)) ⊆ dom (𝑚𝑛))
8882, 83, 85, 87syl3anc 1373 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)) ⊆ dom (𝑚𝑛))
8978, 88eqsstrd 4018 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (Base‘𝑃) ⊆ dom (𝑚𝑛))
9089ex 412 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛) → (Base‘𝑃) ⊆ dom (𝑚𝑛)))
91 simprl 771 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 ∈ (𝑃 RingHom 𝑆))
922, 3rhmf 20485 . . . . . . . . 9 (𝑚 ∈ (𝑃 RingHom 𝑆) → 𝑚:(Base‘𝑃)⟶𝐶)
93 ffn 6736 . . . . . . . . 9 (𝑚:(Base‘𝑃)⟶𝐶𝑚 Fn (Base‘𝑃))
9491, 92, 933syl 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))
962, 3rhmf 20485 . . . . . . . . 9 (𝑛 ∈ (𝑃 RingHom 𝑆) → 𝑛:(Base‘𝑃)⟶𝐶)
97 ffn 6736 . . . . . . . . 9 (𝑛:(Base‘𝑃)⟶𝐶𝑛 Fn (Base‘𝑃))
9895, 96, 973syl 18 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 Fn (Base‘𝑃))
9969adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝐴 ⊆ (Base‘𝑃))
10071adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝑉 ⊆ (Base‘𝑃))
10199, 100unssd 4192 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (ran 𝐴 ∪ ran 𝑉) ⊆ (Base‘𝑃))
102 fnreseql 7068 . . . . . . . 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 (𝑚𝑛)))
10394, 98, 101, 102syl3anc 1373 . . . . . . 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 (𝑚𝑛)))
104 fneqeql2 7067 . . . . . . . 8 ((𝑚 Fn (Base‘𝑃) ∧ 𝑛 Fn (Base‘𝑃)) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚𝑛)))
10594, 98, 104syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚𝑛)))
10690, 103, 1053imtr4d 294 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) → 𝑚 = 𝑛))
10743, 106syl5 34 . . . . 5 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)) ∧ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))) → 𝑚 = 𝑛))
108107ralrimivva 3202 . . . 4 (𝜑 → ∀𝑚 ∈ (𝑃 RingHom 𝑆)∀𝑛 ∈ (𝑃 RingHom 𝑆)(((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)) ∧ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))) → 𝑚 = 𝑛))
109 reseq1 5991 . . . . . 6 (𝑚 = 𝑛 → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)))
110109eqeq1d 2739 . . . . 5 (𝑚 = 𝑛 → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)) ↔ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))))
111110rmo4 3736 . . . 4 (∃*𝑚 ∈ (𝑃 RingHom 𝑆)(𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)) ↔ ∀𝑚 ∈ (𝑃 RingHom 𝑆)∀𝑛 ∈ (𝑃 RingHom 𝑆)(((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)) ∧ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))) → 𝑚 = 𝑛))
112108, 111sylibr 234 . . 3 (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)(𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)))
113 rmoim 3746 . . 3 (∀𝑚 ∈ (𝑃 RingHom 𝑆)(((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))) → (∃*𝑚 ∈ (𝑃 RingHom 𝑆)(𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)) → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺)))
11442, 112, 113sylc 65 . 2 (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
115 reu5 3382 . 2 (∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) ↔ (∃𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) ∧ ∃*𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺)))
11624, 114, 115sylanbrc 583 1 (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  ∃!wreu 3378  ∃*wrmo 3379  {crab 3436  cun 3949  cin 3950  wss 3951  𝒫 cpw 4600  cmpt 5225  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  ccom 5689  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  f cof 7695  m cmap 8866  Fincfn 8985  cn 12266  0cn0 12526  Basecbs 17247  .rcmulr 17298   Σg cgsu 17485  Moorecmre 17625  mrClscmrc 17626  .gcmg 19085  mulGrpcmgp 20137  Ringcrg 20230  CRingccrg 20231   RingHom crh 20469  SubRingcsubrg 20569  AssAlgcasa 21870  AlgSpancasp 21871  algSccascl 21872   mPwSer cmps 21924   mVar cmvr 21925   mPoly cmpl 21926
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-rhm 20472  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-lsp 20970  df-assa 21873  df-asp 21874  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931
This theorem is referenced by:  evlsval2  22111  evlsval3  42569
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