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Theorem evlseu 19438
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 (𝜑𝐼 ∈ V)
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)

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 2621 . . . 4 (Base‘𝑃) = (Base‘𝑃)
3 evlseu.c . . . 4 𝐶 = (Base‘𝑆)
4 eqid 2621 . . . 4 (Base‘𝑅) = (Base‘𝑅)
5 eqid 2621 . . . 4 {𝑧 ∈ (ℕ0𝑚 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} = {𝑧 ∈ (ℕ0𝑚 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin}
6 eqid 2621 . . . 4 (mulGrp‘𝑆) = (mulGrp‘𝑆)
7 eqid 2621 . . . 4 (.g‘(mulGrp‘𝑆)) = (.g‘(mulGrp‘𝑆))
8 eqid 2621 . . . 4 (.r𝑆) = (.r𝑆)
9 evlseu.v . . . 4 𝑉 = (𝐼 mVar 𝑅)
10 eqid 2621 . . . 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))))))
11 evlseu.i . . . 4 (𝜑𝐼 ∈ V)
12 evlseu.r . . . 4 (𝜑𝑅 ∈ CRing)
13 evlseu.s . . . 4 (𝜑𝑆 ∈ CRing)
14 evlseu.f . . . 4 (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))
15 evlseu.g . . . 4 (𝜑𝐺:𝐼𝐶)
16 evlseu.a . . . 4 𝐴 = (algSc‘𝑃)
171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16evlslem1 19437 . . 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))
18 coeq1 5241 . . . . . . 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))
1918eqeq1d 2623 . . . . . 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))
20 coeq1 5241 . . . . . . 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))
2120eqeq1d 2623 . . . . . 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))
2219, 21anbi12d 746 . . . . 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)))
2322rspcev 3295 . . . 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))
24233impb 1257 . . 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))
2517, 24syl 17 . 2 (𝜑 → ∃𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
26 crngring 18482 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2712, 26syl 17 . . . . . . . . . 10 (𝜑𝑅 ∈ Ring)
28 eqid 2621 . . . . . . . . . . 11 (Scalar‘𝑃) = (Scalar‘𝑃)
291mplring 19374 . . . . . . . . . . 11 ((𝐼 ∈ V ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
301mpllmod 19373 . . . . . . . . . . 11 ((𝐼 ∈ V ∧ 𝑅 ∈ Ring) → 𝑃 ∈ LMod)
31 eqid 2621 . . . . . . . . . . 11 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
3216, 28, 29, 30, 31, 2asclf 19259 . . . . . . . . . 10 ((𝐼 ∈ V ∧ 𝑅 ∈ Ring) → 𝐴:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
3311, 27, 32syl2anc 692 . . . . . . . . 9 (𝜑𝐴:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
34 ffun 6007 . . . . . . . . 9 (𝐴:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃) → Fun 𝐴)
3533, 34syl 17 . . . . . . . 8 (𝜑 → Fun 𝐴)
36 funcoeqres 6126 . . . . . . . 8 ((Fun 𝐴 ∧ (𝑚𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹𝐴))
3735, 36sylan 488 . . . . . . 7 ((𝜑 ∧ (𝑚𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹𝐴))
381, 9, 2, 11, 27mvrf2 19414 . . . . . . . . 9 (𝜑𝑉:𝐼⟶(Base‘𝑃))
39 ffun 6007 . . . . . . . . 9 (𝑉:𝐼⟶(Base‘𝑃) → Fun 𝑉)
4038, 39syl 17 . . . . . . . 8 (𝜑 → Fun 𝑉)
41 funcoeqres 6126 . . . . . . . 8 ((Fun 𝑉 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺𝑉))
4240, 41sylan 488 . . . . . . 7 ((𝜑 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺𝑉))
4337, 42anim12dan 881 . . . . . 6 ((𝜑 ∧ ((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺)) → ((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉)))
4443ex 450 . . . . 5 (𝜑 → (((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) → ((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉))))
45 resundi 5371 . . . . . 6 (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉))
46 uneq12 3742 . . . . . 6 (((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉)) → ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)))
4745, 46syl5eq 2667 . . . . 5 (((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉)) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)))
4844, 47syl6 35 . . . 4 (𝜑 → (((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))))
4948ralrimivw 2961 . . 3 (𝜑 → ∀𝑚 ∈ (𝑃 RingHom 𝑆)(((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))))
50 eqtr3 2642 . . . . . 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)))
51 eqid 2621 . . . . . . . . . . . . 13 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
5251, 11, 12psrassa 19336 . . . . . . . . . . . 12 (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg)
53 eqid 2621 . . . . . . . . . . . . . 14 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
5451, 9, 53, 11, 27mvrf 19346 . . . . . . . . . . . . 13 (𝜑𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅)))
55 frn 6012 . . . . . . . . . . . . 13 (𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅)) → ran 𝑉 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
5654, 55syl 17 . . . . . . . . . . . 12 (𝜑 → ran 𝑉 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
57 eqid 2621 . . . . . . . . . . . . 13 (AlgSpan‘(𝐼 mPwSer 𝑅)) = (AlgSpan‘(𝐼 mPwSer 𝑅))
58 eqid 2621 . . . . . . . . . . . . 13 (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘(𝐼 mPwSer 𝑅))
59 eqid 2621 . . . . . . . . . . . . 13 (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅))) = (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))
6057, 58, 59, 53aspval2 19269 . . . . . . . . . . . 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)))
6152, 56, 60syl2anc 692 . . . . . . . . . . 11 (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)))
621, 51, 9, 57, 11, 12mplbas2 19392 . . . . . . . . . . 11 (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑃))
6351, 1, 2, 11, 27mplsubrg 19362 . . . . . . . . . . . . . . 15 (𝜑 → (Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
641, 51, 2mplval2 19353 . . . . . . . . . . . . . . . 16 𝑃 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑃))
6564subsubrg2 18731 . . . . . . . . . . . . . . 15 ((Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → (SubRing‘𝑃) = ((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
6663, 65syl 17 . . . . . . . . . . . . . 14 (𝜑 → (SubRing‘𝑃) = ((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
6766fveq2d 6154 . . . . . . . . . . . . 13 (𝜑 → (mrCls‘(SubRing‘𝑃)) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))))
6858, 64ressascl 19266 . . . . . . . . . . . . . . . . 17 ((Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘𝑃))
6963, 68syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘𝑃))
7069, 16syl6reqr 2674 . . . . . . . . . . . . . . 15 (𝜑𝐴 = (algSc‘(𝐼 mPwSer 𝑅)))
7170rneqd 5315 . . . . . . . . . . . . . 14 (𝜑 → ran 𝐴 = ran (algSc‘(𝐼 mPwSer 𝑅)))
7271uneq1d 3746 . . . . . . . . . . . . 13 (𝜑 → (ran 𝐴 ∪ ran 𝑉) = (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉))
7367, 72fveq12d 6156 . . . . . . . . . . . 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 19242 . . . . . . . . . . . . . 14 ((𝐼 mPwSer 𝑅) ∈ AssAlg → (𝐼 mPwSer 𝑅) ∈ Ring)
7553subrgmre 18728 . . . . . . . . . . . . . 14 ((𝐼 mPwSer 𝑅) ∈ Ring → (SubRing‘(𝐼 mPwSer 𝑅)) ∈ (Moore‘(Base‘(𝐼 mPwSer 𝑅))))
7652, 74, 753syl 18 . . . . . . . . . . . . 13 (𝜑 → (SubRing‘(𝐼 mPwSer 𝑅)) ∈ (Moore‘(Base‘(𝐼 mPwSer 𝑅))))
77 frn 6012 . . . . . . . . . . . . . . . 16 (𝐴:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃) → ran 𝐴 ⊆ (Base‘𝑃))
7833, 77syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐴 ⊆ (Base‘𝑃))
7971, 78eqsstr3d 3621 . . . . . . . . . . . . . 14 (𝜑 → ran (algSc‘(𝐼 mPwSer 𝑅)) ⊆ (Base‘𝑃))
80 frn 6012 . . . . . . . . . . . . . . 15 (𝑉:𝐼⟶(Base‘𝑃) → ran 𝑉 ⊆ (Base‘𝑃))
8138, 80syl 17 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
8279, 81unssd 3769 . . . . . . . . . . . . 13 (𝜑 → (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉) ⊆ (Base‘𝑃))
83 eqid 2621 . . . . . . . . . . . . . 14 (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
8459, 83submrc 16212 . . . . . . . . . . . . 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)))
8576, 63, 82, 84syl3anc 1323 . . . . . . . . . . . 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)))
8673, 85eqtr2d 2656 . . . . . . . . . . 11 (𝜑 → ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
8761, 62, 863eqtr3d 2663 . . . . . . . . . 10 (𝜑 → (Base‘𝑃) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
8887ad2antrr 761 . . . . . . . . 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)))
8911, 27, 29syl2anc 692 . . . . . . . . . . . 12 (𝜑𝑃 ∈ Ring)
902subrgmre 18728 . . . . . . . . . . . 12 (𝑃 ∈ Ring → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
9189, 90syl 17 . . . . . . . . . . 11 (𝜑 → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
9291ad2antrr 761 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
93 simpr 477 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛))
94 rhmeql 18734 . . . . . . . . . . 11 ((𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆)) → dom (𝑚𝑛) ∈ (SubRing‘𝑃))
9594ad2antlr 762 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → dom (𝑚𝑛) ∈ (SubRing‘𝑃))
96 eqid 2621 . . . . . . . . . . 11 (mrCls‘(SubRing‘𝑃)) = (mrCls‘(SubRing‘𝑃))
9796mrcsscl 16204 . . . . . . . . . 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))
9892, 93, 95, 97syl3anc 1323 . . . . . . . . 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))
9988, 98eqsstrd 3620 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (Base‘𝑃) ⊆ dom (𝑚𝑛))
10099ex 450 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛) → (Base‘𝑃) ⊆ dom (𝑚𝑛)))
101 simprl 793 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 ∈ (𝑃 RingHom 𝑆))
1022, 3rhmf 18650 . . . . . . . . 9 (𝑚 ∈ (𝑃 RingHom 𝑆) → 𝑚:(Base‘𝑃)⟶𝐶)
103 ffn 6004 . . . . . . . . 9 (𝑚:(Base‘𝑃)⟶𝐶𝑚 Fn (Base‘𝑃))
104101, 102, 1033syl 18 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 Fn (Base‘𝑃))
105 simprr 795 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 ∈ (𝑃 RingHom 𝑆))
1062, 3rhmf 18650 . . . . . . . . 9 (𝑛 ∈ (𝑃 RingHom 𝑆) → 𝑛:(Base‘𝑃)⟶𝐶)
107 ffn 6004 . . . . . . . . 9 (𝑛:(Base‘𝑃)⟶𝐶𝑛 Fn (Base‘𝑃))
108105, 106, 1073syl 18 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 Fn (Base‘𝑃))
10978adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝐴 ⊆ (Base‘𝑃))
11081adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝑉 ⊆ (Base‘𝑃))
111109, 110unssd 3769 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (ran 𝐴 ∪ ran 𝑉) ⊆ (Base‘𝑃))
112 fnreseql 6285 . . . . . . . 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)))
113104, 108, 111, 112syl3anc 1323 . . . . . . 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)))
114 fneqeql2 6284 . . . . . . . 8 ((𝑚 Fn (Base‘𝑃) ∧ 𝑛 Fn (Base‘𝑃)) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚𝑛)))
115104, 108, 114syl2anc 692 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚𝑛)))
116100, 113, 1153imtr4d 283 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) → 𝑚 = 𝑛))
11750, 116syl5 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))
118117ralrimivva 2965 . . . 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))
119 reseq1 5352 . . . . . 6 (𝑚 = 𝑛 → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)))
120119eqeq1d 2623 . . . . 5 (𝑚 = 𝑛 → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)) ↔ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))))
121120rmo4 3382 . . . 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))
122118, 121sylibr 224 . . 3 (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)(𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)))
123 rmoim 3390 . . 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)))
12449, 122, 123sylc 65 . 2 (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
125 reu5 3148 . 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)))
12625, 124, 125sylanbrc 697 1 (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  ∃!wreu 2909  ∃*wrmo 2910  {crab 2911  Vcvv 3186  cun 3554  cin 3555  wss 3556  𝒫 cpw 4132  cmpt 4675  ccnv 5075  dom cdm 5076  ran crn 5077  cres 5078  cima 5079  ccom 5080  Fun wfun 5843   Fn wfn 5844  wf 5845  cfv 5849  (class class class)co 6607  𝑓 cof 6851  𝑚 cmap 7805  Fincfn 7902  cn 10967  0cn0 11239  Basecbs 15784  .rcmulr 15866  Scalarcsca 15868   Σg cgsu 16025  Moorecmre 16166  mrClscmrc 16167  .gcmg 17464  mulGrpcmgp 18413  Ringcrg 18471  CRingccrg 18472   RingHom crh 18636  SubRingcsubrg 18700  AssAlgcasa 19231  AlgSpancasp 19232  algSccascl 19233   mPwSer cmps 19273   mVar cmvr 19274   mPoly cmpl 19275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-iin 4490  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-of 6853  df-ofr 6854  df-om 7016  df-1st 7116  df-2nd 7117  df-supp 7244  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-map 7807  df-pm 7808  df-ixp 7856  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-fsupp 8223  df-oi 8362  df-card 8712  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-3 11027  df-4 11028  df-5 11029  df-6 11030  df-7 11031  df-8 11032  df-9 11033  df-n0 11240  df-z 11325  df-uz 11635  df-fz 12272  df-fzo 12410  df-seq 12745  df-hash 13061  df-struct 15786  df-ndx 15787  df-slot 15788  df-base 15789  df-sets 15790  df-ress 15791  df-plusg 15878  df-mulr 15879  df-sca 15881  df-vsca 15882  df-tset 15884  df-0g 16026  df-gsum 16027  df-mre 16170  df-mrc 16171  df-acs 16173  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-mhm 17259  df-submnd 17260  df-grp 17349  df-minusg 17350  df-sbg 17351  df-mulg 17465  df-subg 17515  df-ghm 17582  df-cntz 17674  df-cmn 18119  df-abl 18120  df-mgp 18414  df-ur 18426  df-srg 18430  df-ring 18473  df-cring 18474  df-rnghom 18639  df-subrg 18702  df-lmod 18789  df-lss 18855  df-lsp 18894  df-assa 19234  df-asp 19235  df-ascl 19236  df-psr 19278  df-mvr 19279  df-mpl 19280
This theorem is referenced by:  evlsval2  19442
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