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Theorem evlseu 19983
 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 2794 . . . 4 (Base‘𝑃) = (Base‘𝑃)
3 evlseu.c . . . 4 𝐶 = (Base‘𝑆)
4 eqid 2794 . . . 4 (Base‘𝑅) = (Base‘𝑅)
5 eqid 2794 . . . 4 {𝑧 ∈ (ℕ0𝑚 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin} = {𝑧 ∈ (ℕ0𝑚 𝐼) ∣ (𝑧 “ ℕ) ∈ Fin}
6 eqid 2794 . . . 4 (mulGrp‘𝑆) = (mulGrp‘𝑆)
7 eqid 2794 . . . 4 (.g‘(mulGrp‘𝑆)) = (.g‘(mulGrp‘𝑆))
8 eqid 2794 . . . 4 (.r𝑆) = (.r𝑆)
9 evlseu.v . . . 4 𝑉 = (𝐼 mVar 𝑅)
10 eqid 2794 . . . 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 19982 . . 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 5617 . . . . . . 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 2796 . . . . . 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 5617 . . . . . . 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 2796 . . . . . 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 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)))
2322rspcev 3557 . . . 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 1108 . . 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 18998 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2712, 26syl 17 . . . . . . . . . 10 (𝜑𝑅 ∈ Ring)
28 eqid 2794 . . . . . . . . . . 11 (Scalar‘𝑃) = (Scalar‘𝑃)
291mplring 19920 . . . . . . . . . . 11 ((𝐼 ∈ V ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
301mpllmod 19919 . . . . . . . . . . 11 ((𝐼 ∈ V ∧ 𝑅 ∈ Ring) → 𝑃 ∈ LMod)
31 eqid 2794 . . . . . . . . . . 11 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
3216, 28, 29, 30, 31, 2asclf 19799 . . . . . . . . . 10 ((𝐼 ∈ V ∧ 𝑅 ∈ Ring) → 𝐴:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
3311, 27, 32syl2anc 584 . . . . . . . . 9 (𝜑𝐴:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
3433ffund 6389 . . . . . . . 8 (𝜑 → Fun 𝐴)
35 funcoeqres 6516 . . . . . . . 8 ((Fun 𝐴 ∧ (𝑚𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹𝐴))
3634, 35sylan 580 . . . . . . 7 ((𝜑 ∧ (𝑚𝐴) = 𝐹) → (𝑚 ↾ ran 𝐴) = (𝐹𝐴))
371, 9, 2, 11, 27mvrf2 19959 . . . . . . . . 9 (𝜑𝑉:𝐼⟶(Base‘𝑃))
3837ffund 6389 . . . . . . . 8 (𝜑 → Fun 𝑉)
39 funcoeqres 6516 . . . . . . . 8 ((Fun 𝑉 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺𝑉))
4038, 39sylan 580 . . . . . . 7 ((𝜑 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ ran 𝑉) = (𝐺𝑉))
4136, 40anim12dan 618 . . . . . 6 ((𝜑 ∧ ((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺)) → ((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉)))
4241ex 413 . . . . 5 (𝜑 → (((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) → ((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉))))
43 resundi 5751 . . . . . 6 (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉))
44 uneq12 4057 . . . . . 6 (((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉)) → ((𝑚 ↾ ran 𝐴) ∪ (𝑚 ↾ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)))
4543, 44syl5eq 2842 . . . . 5 (((𝑚 ↾ ran 𝐴) = (𝐹𝐴) ∧ (𝑚 ↾ ran 𝑉) = (𝐺𝑉)) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)))
4642, 45syl6 35 . . . 4 (𝜑 → (((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))))
4746ralrimivw 3149 . . 3 (𝜑 → ∀𝑚 ∈ (𝑃 RingHom 𝑆)(((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺) → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))))
48 eqtr3 2817 . . . . . 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)))
49 eqid 2794 . . . . . . . . . . . . 13 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
5049, 11, 12psrassa 19882 . . . . . . . . . . . 12 (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg)
51 eqid 2794 . . . . . . . . . . . . . 14 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
5249, 9, 51, 11, 27mvrf 19892 . . . . . . . . . . . . 13 (𝜑𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅)))
5352frnd 6392 . . . . . . . . . . . 12 (𝜑 → ran 𝑉 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
54 eqid 2794 . . . . . . . . . . . . 13 (AlgSpan‘(𝐼 mPwSer 𝑅)) = (AlgSpan‘(𝐼 mPwSer 𝑅))
55 eqid 2794 . . . . . . . . . . . . 13 (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘(𝐼 mPwSer 𝑅))
56 eqid 2794 . . . . . . . . . . . . 13 (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅))) = (mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))
5754, 55, 56, 51aspval2 19815 . . . . . . . . . . . 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)))
5850, 53, 57syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)))
591, 49, 9, 54, 11, 12mplbas2 19938 . . . . . . . . . . 11 (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑃))
6049, 1, 2, 11, 27mplsubrg 19908 . . . . . . . . . . . . . . 15 (𝜑 → (Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
611, 49, 2mplval2 19899 . . . . . . . . . . . . . . . 16 𝑃 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑃))
6261subsubrg2 19252 . . . . . . . . . . . . . . 15 ((Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → (SubRing‘𝑃) = ((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
6360, 62syl 17 . . . . . . . . . . . . . 14 (𝜑 → (SubRing‘𝑃) = ((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
6463fveq2d 6545 . . . . . . . . . . . . 13 (𝜑 → (mrCls‘(SubRing‘𝑃)) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))))
6555, 61ressascl 19812 . . . . . . . . . . . . . . . . 17 ((Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘𝑃))
6660, 65syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (algSc‘(𝐼 mPwSer 𝑅)) = (algSc‘𝑃))
6766, 16syl6reqr 2849 . . . . . . . . . . . . . . 15 (𝜑𝐴 = (algSc‘(𝐼 mPwSer 𝑅)))
6867rneqd 5693 . . . . . . . . . . . . . 14 (𝜑 → ran 𝐴 = ran (algSc‘(𝐼 mPwSer 𝑅)))
6968uneq1d 4061 . . . . . . . . . . . . 13 (𝜑 → (ran 𝐴 ∪ ran 𝑉) = (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉))
7064, 69fveq12d 6548 . . . . . . . . . . . 12 (𝜑 → ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)) = ((mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)))
71 assaring 19782 . . . . . . . . . . . . . 14 ((𝐼 mPwSer 𝑅) ∈ AssAlg → (𝐼 mPwSer 𝑅) ∈ Ring)
7251subrgmre 19249 . . . . . . . . . . . . . 14 ((𝐼 mPwSer 𝑅) ∈ Ring → (SubRing‘(𝐼 mPwSer 𝑅)) ∈ (Moore‘(Base‘(𝐼 mPwSer 𝑅))))
7350, 71, 723syl 18 . . . . . . . . . . . . 13 (𝜑 → (SubRing‘(𝐼 mPwSer 𝑅)) ∈ (Moore‘(Base‘(𝐼 mPwSer 𝑅))))
7433frnd 6392 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐴 ⊆ (Base‘𝑃))
7568, 74eqsstrrd 3929 . . . . . . . . . . . . . 14 (𝜑 → ran (algSc‘(𝐼 mPwSer 𝑅)) ⊆ (Base‘𝑃))
7637frnd 6392 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
7775, 76unssd 4085 . . . . . . . . . . . . 13 (𝜑 → (ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉) ⊆ (Base‘𝑃))
78 eqid 2794 . . . . . . . . . . . . . 14 (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃))) = (mrCls‘((SubRing‘(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Base‘𝑃)))
7956, 78submrc 16728 . . . . . . . . . . . . 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)))
8073, 60, 77, 79syl3anc 1364 . . . . . . . . . . . 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)))
8170, 80eqtr2d 2831 . . . . . . . . . . 11 (𝜑 → ((mrCls‘(SubRing‘(𝐼 mPwSer 𝑅)))‘(ran (algSc‘(𝐼 mPwSer 𝑅)) ∪ ran 𝑉)) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
8258, 59, 813eqtr3d 2838 . . . . . . . . . 10 (𝜑 → (Base‘𝑃) = ((mrCls‘(SubRing‘𝑃))‘(ran 𝐴 ∪ ran 𝑉)))
8382ad2antrr 722 . . . . . . . . 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)))
8411, 27, 29syl2anc 584 . . . . . . . . . . . 12 (𝜑𝑃 ∈ Ring)
852subrgmre 19249 . . . . . . . . . . . 12 (𝑃 ∈ Ring → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
8684, 85syl 17 . . . . . . . . . . 11 (𝜑 → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
8786ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (SubRing‘𝑃) ∈ (Moore‘(Base‘𝑃)))
88 simpr 485 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛))
89 rhmeql 19255 . . . . . . . . . . 11 ((𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆)) → dom (𝑚𝑛) ∈ (SubRing‘𝑃))
9089ad2antlr 723 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → dom (𝑚𝑛) ∈ (SubRing‘𝑃))
91 eqid 2794 . . . . . . . . . . 11 (mrCls‘(SubRing‘𝑃)) = (mrCls‘(SubRing‘𝑃))
9291mrcsscl 16720 . . . . . . . . . 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))
9387, 88, 90, 92syl3anc 1364 . . . . . . . . 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))
9483, 93eqsstrd 3928 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛)) → (Base‘𝑃) ⊆ dom (𝑚𝑛))
9594ex 413 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((ran 𝐴 ∪ ran 𝑉) ⊆ dom (𝑚𝑛) → (Base‘𝑃) ⊆ dom (𝑚𝑛)))
96 simprl 767 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 ∈ (𝑃 RingHom 𝑆))
972, 3rhmf 19168 . . . . . . . . 9 (𝑚 ∈ (𝑃 RingHom 𝑆) → 𝑚:(Base‘𝑃)⟶𝐶)
98 ffn 6385 . . . . . . . . 9 (𝑚:(Base‘𝑃)⟶𝐶𝑚 Fn (Base‘𝑃))
9996, 97, 983syl 18 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑚 Fn (Base‘𝑃))
100 simprr 769 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 ∈ (𝑃 RingHom 𝑆))
1012, 3rhmf 19168 . . . . . . . . 9 (𝑛 ∈ (𝑃 RingHom 𝑆) → 𝑛:(Base‘𝑃)⟶𝐶)
102 ffn 6385 . . . . . . . . 9 (𝑛:(Base‘𝑃)⟶𝐶𝑛 Fn (Base‘𝑃))
103100, 101, 1023syl 18 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → 𝑛 Fn (Base‘𝑃))
10474adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝐴 ⊆ (Base‘𝑃))
10576adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ran 𝑉 ⊆ (Base‘𝑃))
106104, 105unssd 4085 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (ran 𝐴 ∪ ran 𝑉) ⊆ (Base‘𝑃))
107 fnreseql 6686 . . . . . . . 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)))
10899, 103, 106, 107syl3anc 1364 . . . . . . 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)))
109 fneqeql2 6685 . . . . . . . 8 ((𝑚 Fn (Base‘𝑃) ∧ 𝑛 Fn (Base‘𝑃)) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚𝑛)))
11099, 103, 109syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → (𝑚 = 𝑛 ↔ (Base‘𝑃) ⊆ dom (𝑚𝑛)))
11195, 108, 1103imtr4d 295 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) → 𝑚 = 𝑛))
11248, 111syl5 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))
113112ralrimivva 3157 . . . 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))
114 reseq1 5731 . . . . . 6 (𝑚 = 𝑛 → (𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)))
115114eqeq1d 2796 . . . . 5 (𝑚 = 𝑛 → ((𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)) ↔ (𝑛 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉))))
116115rmo4 3656 . . . 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))
117113, 116sylibr 235 . . 3 (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)(𝑚 ↾ (ran 𝐴 ∪ ran 𝑉)) = ((𝐹𝐴) ∪ (𝐺𝑉)))
118 rmoim 3666 . . 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)))
11947, 117, 118sylc 65 . 2 (𝜑 → ∃*𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
120 reu5 3389 . 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)))
12125, 119, 120sylanbrc 583 1 (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2080  ∀wral 3104  ∃wrex 3105  ∃!wreu 3106  ∃*wrmo 3107  {crab 3108  Vcvv 3436   ∪ cun 3859   ∩ cin 3860   ⊆ wss 3861  𝒫 cpw 4455   ↦ cmpt 5043  ◡ccnv 5445  dom cdm 5446  ran crn 5447   ↾ cres 5448   “ cima 5449   ∘ ccom 5450  Fun wfun 6222   Fn wfn 6223  ⟶wf 6224  ‘cfv 6228  (class class class)co 7019   ∘𝑓 cof 7268   ↑𝑚 cmap 8259  Fincfn 8360  ℕcn 11488  ℕ0cn0 11747  Basecbs 16312  .rcmulr 16395  Scalarcsca 16397   Σg cgsu 16543  Moorecmre 16682  mrClscmrc 16683  .gcmg 17981  mulGrpcmgp 18929  Ringcrg 18987  CRingccrg 18988   RingHom crh 19154  SubRingcsubrg 19221  AssAlgcasa 19771  AlgSpancasp 19772  algSccascl 19773   mPwSer cmps 19819   mVar cmvr 19820   mPoly cmpl 19821 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322  ax-cnex 10442  ax-resscn 10443  ax-1cn 10444  ax-icn 10445  ax-addcl 10446  ax-addrcl 10447  ax-mulcl 10448  ax-mulrcl 10449  ax-mulcom 10450  ax-addass 10451  ax-mulass 10452  ax-distr 10453  ax-i2m1 10454  ax-1ne0 10455  ax-1rid 10456  ax-rnegex 10457  ax-rrecex 10458  ax-cnre 10459  ax-pre-lttri 10460  ax-pre-lttrn 10461  ax-pre-ltadd 10462  ax-pre-mulgt0 10463 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-nel 3090  df-ral 3109  df-rex 3110  df-reu 3111  df-rmo 3112  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-int 4785  df-iun 4829  df-iin 4830  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-se 5406  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-isom 6237  df-riota 6980  df-ov 7022  df-oprab 7023  df-mpo 7024  df-of 7270  df-ofr 7271  df-om 7440  df-1st 7548  df-2nd 7549  df-supp 7685  df-wrecs 7801  df-recs 7863  df-rdg 7901  df-1o 7956  df-2o 7957  df-oadd 7960  df-er 8142  df-map 8261  df-pm 8262  df-ixp 8314  df-en 8361  df-dom 8362  df-sdom 8363  df-fin 8364  df-fsupp 8683  df-oi 8823  df-card 9217  df-pnf 10526  df-mnf 10527  df-xr 10528  df-ltxr 10529  df-le 10530  df-sub 10721  df-neg 10722  df-nn 11489  df-2 11550  df-3 11551  df-4 11552  df-5 11553  df-6 11554  df-7 11555  df-8 11556  df-9 11557  df-n0 11748  df-z 11832  df-uz 12094  df-fz 12743  df-fzo 12884  df-seq 13220  df-hash 13541  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-mulr 16408  df-sca 16410  df-vsca 16411  df-tset 16413  df-0g 16544  df-gsum 16545  df-mre 16686  df-mrc 16687  df-acs 16689  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-mhm 17774  df-submnd 17775  df-grp 17864  df-minusg 17865  df-sbg 17866  df-mulg 17982  df-subg 18030  df-ghm 18097  df-cntz 18188  df-cmn 18635  df-abl 18636  df-mgp 18930  df-ur 18942  df-srg 18946  df-ring 18989  df-cring 18990  df-rnghom 19157  df-subrg 19223  df-lmod 19326  df-lss 19394  df-lsp 19434  df-assa 19774  df-asp 19775  df-ascl 19776  df-psr 19824  df-mvr 19825  df-mpl 19826 This theorem is referenced by:  evlsval2  19987
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