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Theorem evlseu 19337
 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 2610 . . . 4 (Base‘𝑃) = (Base‘𝑃)
3 evlseu.c . . . 4 𝐶 = (Base‘𝑆)
4 eqid 2610 . . . 4 (Base‘𝑅) = (Base‘𝑅)
5 eqid 2610 . . . 4 {𝑧 ∈ (ℕ0𝑚 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} = {𝑧 ∈ (ℕ0𝑚 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin}
6 eqid 2610 . . . 4 (mulGrp‘𝑆) = (mulGrp‘𝑆)
7 eqid 2610 . . . 4 (.g‘(mulGrp‘𝑆)) = (.g‘(mulGrp‘𝑆))
8 eqid 2610 . . . 4 (.r𝑆) = (.r𝑆)
9 evlseu.v . . . 4 𝑉 = (𝐼 mVar 𝑅)
10 eqid 2610 . . . 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 19336 . . 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 5201 . . . . . . 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 2612 . . . . . 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 5201 . . . . . . 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 2612 . . . . . 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 743 . . . . 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 3282 . . . 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 1252 . . 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 18381 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2712, 26syl 17 . . . . . . . . . 10 (𝜑𝑅 ∈ Ring)
28 eqid 2610 . . . . . . . . . . 11 (Scalar‘𝑃) = (Scalar‘𝑃)
291mplring 19273 . . . . . . . . . . 11 ((𝐼 ∈ V ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
301mpllmod 19272 . . . . . . . . . . 11 ((𝐼 ∈ V ∧ 𝑅 ∈ Ring) → 𝑃 ∈ LMod)
31 eqid 2610 . . . . . . . . . . 11 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
3216, 28, 29, 30, 31, 2asclf 19158 . . . . . . . . . 10 ((𝐼 ∈ V ∧ 𝑅 ∈ Ring) → 𝐴:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
3311, 27, 32syl2anc 691 . . . . . . . . 9 (𝜑𝐴:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
34 ffun 5961 . . . . . . . . 9 (𝐴:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃) → Fun 𝐴)
3533, 34syl 17 . . . . . . . 8 (𝜑 → Fun 𝐴)
36 funcoeqres 6080 . . . . . . . 8 ((Fun 𝐴 ∧ (𝑚𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹𝐴))
3735, 36sylan 487 . . . . . . 7 ((𝜑 ∧ (𝑚𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹𝐴))
381, 9, 2, 11, 27mvrf2 19313 . . . . . . . . 9 (𝜑𝑉:𝐼⟶(Base‘𝑃))
39 ffun 5961 . . . . . . . . 9 (𝑉:𝐼⟶(Base‘𝑃) → Fun 𝑉)
4038, 39syl 17 . . . . . . . 8 (𝜑 → Fun 𝑉)
41 funcoeqres 6080 . . . . . . . 8 ((Fun 𝑉 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺𝑉))
4240, 41sylan 487 . . . . . . 7 ((𝜑 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺𝑉))
4337, 42anim12dan 878 . . . . . 6 ((𝜑 ∧ ((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺)) → ((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉)))
4443ex 449 . . . . 5 (𝜑 → (((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) → ((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉))))
45 resundi 5330 . . . . . 6 (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉))
46 uneq12 3724 . . . . . 6 (((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉)) → ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)))
4745, 46syl5eq 2656 . . . . 5 (((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉)) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)))
4844, 47syl6 34 . . . 4 (𝜑 → (((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))))
4948ralrimivw 2950 . . 3 (𝜑 → ∀𝑚 ∈ (𝑃 RingHom 𝑆)(((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))))
50 eqtr3 2631 . . . . . 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 2610 . . . . . . . . . . . . 13 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
5251, 11, 12psrassa 19235 . . . . . . . . . . . 12 (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg)
53 eqid 2610 . . . . . . . . . . . . . 14 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
5451, 9, 53, 11, 27mvrf 19245 . . . . . . . . . . . . 13 (𝜑𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅)))
55 frn 5966 . . . . . . . . . . . . 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 2610 . . . . . . . . . . . . 13 (AlgSpan‘(𝐼 mPwSer 𝑅)) = (AlgSpan‘(𝐼 mPwSer 𝑅))
58 eqid 2610 . . . . . . . . . . . . 13 (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘(𝐼 mPwSer 𝑅))
59 eqid 2610 . . . . . . . . . . . . 13 (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅))) = (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))
6057, 58, 59, 53aspval2 19168 . . . . . . . . . . . 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 691 . . . . . . . . . . 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 19291 . . . . . . . . . . 11 (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑃))
6351, 1, 2, 11, 27mplsubrg 19261 . . . . . . . . . . . . . . 15 (𝜑 → (Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
641, 51, 2mplval2 19252 . . . . . . . . . . . . . . . 16 𝑃 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑃))
6564subsubrg2 18630 . . . . . . . . . . . . . . 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 6107 . . . . . . . . . . . . 13 (𝜑 → (mrCls‘(SubRing‘𝑃)) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))))
6858, 64ressascl 19165 . . . . . . . . . . . . . . . . 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 2663 . . . . . . . . . . . . . . 15 (𝜑𝐴 = (algSc‘(𝐼 mPwSer 𝑅)))
7170rneqd 5274 . . . . . . . . . . . . . 14 (𝜑 → ran 𝐴 = ran (algSc‘(𝐼 mPwSer 𝑅)))
7271uneq1d 3728 . . . . . . . . . . . . 13 (𝜑 → (ran 𝐴 ∪ ran 𝑉) = (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉))
7367, 72fveq12d 6109 . . . . . . . . . . . 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 19141 . . . . . . . . . . . . . 14 ((𝐼 mPwSer 𝑅) ∈ AssAlg → (𝐼 mPwSer 𝑅) ∈ Ring)
7553subrgmre 18627 . . . . . . . . . . . . . 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 5966 . . . . . . . . . . . . . . . 16 (𝐴:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃) → ran 𝐴 ⊆ (Base‘𝑃))
7833, 77syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐴 ⊆ (Base‘𝑃))
7971, 78eqsstr3d 3603 . . . . . . . . . . . . . 14 (𝜑 → ran (algSc‘(𝐼 mPwSer 𝑅)) ⊆ (Base‘𝑃))
80 frn 5966 . . . . . . . . . . . . . . 15 (𝑉:𝐼⟶(Base‘𝑃) → ran 𝑉 ⊆ (Base‘𝑃))
8138, 80syl 17 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
8279, 81unssd 3751 . . . . . . . . . . . . 13 (𝜑 → (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉) ⊆ (Base‘𝑃))
83 eqid 2610 . . . . . . . . . . . . . 14 (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
8459, 83submrc 16111 . . . . . . . . . . . . 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 1318 . . . . . . . . . . . 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 2645 . . . . . . . . . . 11 (𝜑 → ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
8761, 62, 863eqtr3d 2652 . . . . . . . . . 10 (𝜑 → (Base‘𝑃) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
8887ad2antrr 758 . . . . . . . . 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 691 . . . . . . . . . . . 12 (𝜑𝑃 ∈ Ring)
902subrgmre 18627 . . . . . . . . . . . 12 (𝑃 ∈ Ring → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
9189, 90syl 17 . . . . . . . . . . 11 (𝜑 → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
9291ad2antrr 758 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
93 simpr 476 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛))
94 rhmeql 18633 . . . . . . . . . . 11 ((𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆)) → dom (𝑚𝑛) ∈ (SubRing‘𝑃))
9594ad2antlr 759 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → dom (𝑚𝑛) ∈ (SubRing‘𝑃))
96 eqid 2610 . . . . . . . . . . 11 (mrCls‘(SubRing‘𝑃)) = (mrCls‘(SubRing‘𝑃))
9796mrcsscl 16103 . . . . . . . . . 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 1318 . . . . . . . . 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 3602 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (Base‘𝑃) ⊆ dom (𝑚𝑛))
10099ex 449 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛) → (Base‘𝑃) ⊆ dom (𝑚𝑛)))
101 simprl 790 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 ∈ (𝑃 RingHom 𝑆))
1022, 3rhmf 18549 . . . . . . . . 9 (𝑚 ∈ (𝑃 RingHom 𝑆) → 𝑚:(Base‘𝑃)⟶𝐶)
103 ffn 5958 . . . . . . . . 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 792 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 ∈ (𝑃 RingHom 𝑆))
1062, 3rhmf 18549 . . . . . . . . 9 (𝑛 ∈ (𝑃 RingHom 𝑆) → 𝑛:(Base‘𝑃)⟶𝐶)
107 ffn 5958 . . . . . . . . 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 480 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝐴 ⊆ (Base‘𝑃))
11081adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝑉 ⊆ (Base‘𝑃))
111109, 110unssd 3751 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (ran 𝐴 ∪ ran 𝑉) ⊆ (Base‘𝑃))
112 fnreseql 6235 . . . . . . . 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 1318 . . . . . . 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 6234 . . . . . . . 8 ((𝑚 Fn (Base‘𝑃) ∧ 𝑛 Fn (Base‘𝑃)) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚𝑛)))
115104, 108, 114syl2anc 691 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚𝑛)))
116100, 113, 1153imtr4d 282 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) → 𝑚 = 𝑛))
11750, 116syl5 33 . . . . 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 2954 . . . 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 5311 . . . . . 6 (𝑚 = 𝑛 → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)))
120119eqeq1d 2612 . . . . 5 (𝑚 = 𝑛 → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)) ↔ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))))
121120rmo4 3366 . . . 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 223 . . 3 (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)(𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)))
123 rmoim 3374 . . 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 63 . 2 (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
125 reu5 3136 . 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 695 1 (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  ∃*wrmo 2899  {crab 2900  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793   ↑𝑚 cmap 7744  Fincfn 7841  ℕcn 10897  ℕ0cn0 11169  Basecbs 15695  .rcmulr 15769  Scalarcsca 15771   Σg cgsu 15924  Moorecmre 16065  mrClscmrc 16066  .gcmg 17363  mulGrpcmgp 18312  Ringcrg 18370  CRingccrg 18371   RingHom crh 18535  SubRingcsubrg 18599  AssAlgcasa 19130  AlgSpancasp 19131  algSccascl 19132   mPwSer cmps 19172   mVar cmvr 19173   mPoly cmpl 19174 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-srg 18329  df-ring 18372  df-cring 18373  df-rnghom 18538  df-subrg 18601  df-lmod 18688  df-lss 18754  df-lsp 18793  df-assa 19133  df-asp 19134  df-ascl 19135  df-psr 19177  df-mvr 19178  df-mpl 19179 This theorem is referenced by:  evlsval2  19341
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