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Theorem evlseu 21957
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 2731 . . . 4 (Base‘𝑃) = (Base‘𝑃)
3 evlseu.c . . . 4 𝐶 = (Base‘𝑆)
4 eqid 2731 . . . 4 {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} = {𝑧 ∈ (ℕ0m 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin}
5 eqid 2731 . . . 4 (mulGrp‘𝑆) = (mulGrp‘𝑆)
6 eqid 2731 . . . 4 (.g‘(mulGrp‘𝑆)) = (.g‘(mulGrp‘𝑆))
7 eqid 2731 . . . 4 (.r𝑆) = (.r𝑆)
8 evlseu.v . . . 4 𝑉 = (𝐼 mVar 𝑅)
9 eqid 2731 . . . 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 21956 . . 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 5857 . . . . . . 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 2733 . . . . . 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 5857 . . . . . . 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 2733 . . . . . 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 630 . . . . 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 3612 . . . 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 1114 . . 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 2731 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
26 crngring 20146 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2711, 26syl 17 . . . . . . . . . 10 (𝜑𝑅 ∈ Ring)
281, 2, 25, 15, 10, 27mplasclf 21937 . . . . . . . . 9 (𝜑𝐴:(Base‘𝑅)⟶(Base‘𝑃))
2928ffund 6721 . . . . . . . 8 (𝜑 → Fun 𝐴)
30 funcoeqres 6864 . . . . . . . 8 ((Fun 𝐴 ∧ (𝑚𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹𝐴))
3129, 30sylan 579 . . . . . . 7 ((𝜑 ∧ (𝑚𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹𝐴))
321, 8, 2, 10, 27mvrf2 21863 . . . . . . . . 9 (𝜑𝑉:𝐼⟶(Base‘𝑃))
3332ffund 6721 . . . . . . . 8 (𝜑 → Fun 𝑉)
34 funcoeqres 6864 . . . . . . . 8 ((Fun 𝑉 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺𝑉))
3533, 34sylan 579 . . . . . . 7 ((𝜑 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺𝑉))
3631, 35anim12dan 618 . . . . . 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 5995 . . . . . 6 (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉))
39 uneq12 4158 . . . . . 6 (((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉)) → ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)))
4038, 39eqtrid 2783 . . . . 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 3149 . . 3 (𝜑 → ∀𝑚 ∈ (𝑃 RingHom 𝑆)(((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))))
43 eqtr3 2757 . . . . . 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 2731 . . . . . . . . . . . . 13 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
4544, 10, 11psrassa 21845 . . . . . . . . . . . 12 (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg)
46 eqid 2731 . . . . . . . . . . . . . 14 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
4744, 8, 46, 10, 27mvrf 21855 . . . . . . . . . . . . 13 (𝜑𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅)))
4847frnd 6725 . . . . . . . . . . . 12 (𝜑 → ran 𝑉 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
49 eqid 2731 . . . . . . . . . . . . 13 (AlgSpan‘(𝐼 mPwSer 𝑅)) = (AlgSpan‘(𝐼 mPwSer 𝑅))
50 eqid 2731 . . . . . . . . . . . . 13 (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘(𝐼 mPwSer 𝑅))
51 eqid 2731 . . . . . . . . . . . . 13 (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅))) = (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))
5249, 50, 51, 46aspval2 21762 . . . . . . . . . . . 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 583 . . . . . . . . . . 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 21908 . . . . . . . . . . 11 (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑃))
5544, 1, 2, 10, 27mplsubrg 21875 . . . . . . . . . . . . . . 15 (𝜑 → (Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
561, 44, 2mplval2 21866 . . . . . . . . . . . . . . . 16 𝑃 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑃))
5756subsubrg2 20497 . . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . . 13 (𝜑 → (mrCls‘(SubRing‘𝑃)) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))))
6050, 56ressascl 21760 . . . . . . . . . . . . . . . . 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 2790 . . . . . . . . . . . . . . 15 (𝜑𝐴 = (algSc‘(𝐼 mPwSer 𝑅)))
6362rneqd 5937 . . . . . . . . . . . . . 14 (𝜑 → ran 𝐴 = ran (algSc‘(𝐼 mPwSer 𝑅)))
6463uneq1d 4162 . . . . . . . . . . . . 13 (𝜑 → (ran 𝐴 ∪ ran 𝑉) = (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉))
6559, 64fveq12d 6898 . . . . . . . . . . . 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 21726 . . . . . . . . . . . . . 14 ((𝐼 mPwSer 𝑅) ∈ AssAlg → (𝐼 mPwSer 𝑅) ∈ Ring)
6746subrgmre 20495 . . . . . . . . . . . . . 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 6725 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐴 ⊆ (Base‘𝑃))
7063, 69eqsstrrd 4021 . . . . . . . . . . . . . 14 (𝜑 → ran (algSc‘(𝐼 mPwSer 𝑅)) ⊆ (Base‘𝑃))
7132frnd 6725 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
7270, 71unssd 4186 . . . . . . . . . . . . 13 (𝜑 → (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉) ⊆ (Base‘𝑃))
73 eqid 2731 . . . . . . . . . . . . . 14 (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
7451, 73submrc 17579 . . . . . . . . . . . . 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 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)))
7665, 75eqtr2d 2772 . . . . . . . . . . 11 (𝜑 → ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
7753, 54, 763eqtr3d 2779 . . . . . . . . . 10 (𝜑 → (Base‘𝑃) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
7877ad2antrr 723 . . . . . . . . 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)))
791mplring 21889 . . . . . . . . . . . . 13 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
8010, 27, 79syl2anc 583 . . . . . . . . . . . 12 (𝜑𝑃 ∈ Ring)
812subrgmre 20495 . . . . . . . . . . . 12 (𝑃 ∈ Ring → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
8280, 81syl 17 . . . . . . . . . . 11 (𝜑 → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
8382ad2antrr 723 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
84 simpr 484 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛))
85 rhmeql 20501 . . . . . . . . . . 11 ((𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆)) → dom (𝑚𝑛) ∈ (SubRing‘𝑃))
8685ad2antlr 724 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → dom (𝑚𝑛) ∈ (SubRing‘𝑃))
87 eqid 2731 . . . . . . . . . . 11 (mrCls‘(SubRing‘𝑃)) = (mrCls‘(SubRing‘𝑃))
8887mrcsscl 17571 . . . . . . . . . 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))
8983, 84, 86, 88syl3anc 1370 . . . . . . . . 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))
9078, 89eqsstrd 4020 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (Base‘𝑃) ⊆ dom (𝑚𝑛))
9190ex 412 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛) → (Base‘𝑃) ⊆ dom (𝑚𝑛)))
92 simprl 768 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 ∈ (𝑃 RingHom 𝑆))
932, 3rhmf 20383 . . . . . . . . 9 (𝑚 ∈ (𝑃 RingHom 𝑆) → 𝑚:(Base‘𝑃)⟶𝐶)
94 ffn 6717 . . . . . . . . 9 (𝑚:(Base‘𝑃)⟶𝐶𝑚 Fn (Base‘𝑃))
9592, 93, 943syl 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))
972, 3rhmf 20383 . . . . . . . . 9 (𝑛 ∈ (𝑃 RingHom 𝑆) → 𝑛:(Base‘𝑃)⟶𝐶)
98 ffn 6717 . . . . . . . . 9 (𝑛:(Base‘𝑃)⟶𝐶𝑛 Fn (Base‘𝑃))
9996, 97, 983syl 18 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 Fn (Base‘𝑃))
10069adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝐴 ⊆ (Base‘𝑃))
10171adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝑉 ⊆ (Base‘𝑃))
102100, 101unssd 4186 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (ran 𝐴 ∪ ran 𝑉) ⊆ (Base‘𝑃))
103 fnreseql 7049 . . . . . . . 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)))
10495, 99, 102, 103syl3anc 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 (𝑚𝑛)))
105 fneqeql2 7048 . . . . . . . 8 ((𝑚 Fn (Base‘𝑃) ∧ 𝑛 Fn (Base‘𝑃)) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚𝑛)))
10695, 99, 105syl2anc 583 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚𝑛)))
10791, 104, 1063imtr4d 294 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) → 𝑚 = 𝑛))
10843, 107syl5 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))
109108ralrimivva 3199 . . . 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))
110 reseq1 5975 . . . . . 6 (𝑚 = 𝑛 → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)))
111110eqeq1d 2733 . . . . 5 (𝑚 = 𝑛 → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)) ↔ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))))
112111rmo4 3726 . . . 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))
113109, 112sylibr 233 . . 3 (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)(𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)))
114 rmoim 3736 . . 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)))
11542, 113, 114sylc 65 . 2 (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
116 reu5 3377 . 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)))
11724, 115, 116sylanbrc 582 1 (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  wrex 3069  ∃!wreu 3373  ∃*wrmo 3374  {crab 3431  cun 3946  cin 3947  wss 3948  𝒫 cpw 4602  cmpt 5231  ccnv 5675  dom cdm 5676  ran crn 5677  cres 5678  cima 5679  ccom 5680  Fun wfun 6537   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7412  f cof 7672  m cmap 8826  Fincfn 8945  cn 12219  0cn0 12479  Basecbs 17151  .rcmulr 17205   Σg cgsu 17393  Moorecmre 17533  mrClscmrc 17534  .gcmg 18993  mulGrpcmgp 20035  Ringcrg 20134  CRingccrg 20135   RingHom crh 20367  SubRingcsubrg 20465  AssAlgcasa 21715  AlgSpancasp 21716  algSccascl 21717   mPwSer cmps 21767   mVar cmvr 21768   mPoly cmpl 21769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-ofr 7675  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8152  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-map 8828  df-pm 8829  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9368  df-sup 9443  df-oi 9511  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-fz 13492  df-fzo 13635  df-seq 13974  df-hash 14298  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-hom 17228  df-cco 17229  df-0g 17394  df-gsum 17395  df-prds 17400  df-pws 17402  df-mre 17537  df-mrc 17538  df-acs 17540  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-mhm 18711  df-submnd 18712  df-grp 18864  df-minusg 18865  df-sbg 18866  df-mulg 18994  df-subg 19046  df-ghm 19135  df-cntz 19229  df-cmn 19698  df-abl 19699  df-mgp 20036  df-rng 20054  df-ur 20083  df-srg 20088  df-ring 20136  df-cring 20137  df-rhm 20370  df-subrng 20442  df-subrg 20467  df-lmod 20704  df-lss 20775  df-lsp 20815  df-assa 21718  df-asp 21719  df-ascl 21720  df-psr 21772  df-mvr 21773  df-mpl 21774
This theorem is referenced by:  evlsval2  21961  evlsval3  41594
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