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Theorem evlseu 21646
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 𝐢 = (Baseβ€˜π‘†)
evlseu.a 𝐴 = (algScβ€˜π‘ƒ)
evlseu.v 𝑉 = (𝐼 mVar 𝑅)
evlseu.i (πœ‘ β†’ 𝐼 ∈ π‘Š)
evlseu.r (πœ‘ β†’ 𝑅 ∈ CRing)
evlseu.s (πœ‘ β†’ 𝑆 ∈ CRing)
evlseu.f (πœ‘ β†’ 𝐹 ∈ (𝑅 RingHom 𝑆))
evlseu.g (πœ‘ β†’ 𝐺:𝐼⟢𝐢)
Assertion
Ref Expression
evlseu (πœ‘ β†’ βˆƒ!π‘š ∈ (𝑃 RingHom 𝑆)((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺))
Distinct variable groups:   𝐴,π‘š   π‘š,𝐹   π‘š,𝐺   π‘š,𝐼   𝑃,π‘š   πœ‘,π‘š   𝑆,π‘š   π‘š,𝑉
Allowed substitution hints:   𝐢(π‘š)   𝑅(π‘š)   π‘Š(π‘š)

Proof of Theorem evlseu
Dummy variables 𝑛 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlseu.p . . . 4 𝑃 = (𝐼 mPoly 𝑅)
2 eqid 2733 . . . 4 (Baseβ€˜π‘ƒ) = (Baseβ€˜π‘ƒ)
3 evlseu.c . . . 4 𝐢 = (Baseβ€˜π‘†)
4 eqid 2733 . . . 4 {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} = {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin}
5 eqid 2733 . . . 4 (mulGrpβ€˜π‘†) = (mulGrpβ€˜π‘†)
6 eqid 2733 . . . 4 (.gβ€˜(mulGrpβ€˜π‘†)) = (.gβ€˜(mulGrpβ€˜π‘†))
7 eqid 2733 . . . 4 (.rβ€˜π‘†) = (.rβ€˜π‘†)
8 evlseu.v . . . 4 𝑉 = (𝐼 mVar 𝑅)
9 eqid 2733 . . . 4 (π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) = (π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺))))))
10 evlseu.i . . . 4 (πœ‘ β†’ 𝐼 ∈ π‘Š)
11 evlseu.r . . . 4 (πœ‘ β†’ 𝑅 ∈ CRing)
12 evlseu.s . . . 4 (πœ‘ β†’ 𝑆 ∈ CRing)
13 evlseu.f . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝑅 RingHom 𝑆))
14 evlseu.g . . . 4 (πœ‘ β†’ 𝐺:𝐼⟢𝐢)
15 evlseu.a . . . 4 𝐴 = (algScβ€˜π‘ƒ)
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15evlslem1 21645 . . 3 (πœ‘ β†’ ((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∈ (𝑃 RingHom 𝑆) ∧ ((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∘ 𝑉) = 𝐺))
17 coeq1 5858 . . . . . . 7 (π‘š = (π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) β†’ (π‘š ∘ 𝐴) = ((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∘ 𝐴))
1817eqeq1d 2735 . . . . . 6 (π‘š = (π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) β†’ ((π‘š ∘ 𝐴) = 𝐹 ↔ ((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∘ 𝐴) = 𝐹))
19 coeq1 5858 . . . . . . 7 (π‘š = (π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) β†’ (π‘š ∘ 𝑉) = ((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∘ 𝑉))
2019eqeq1d 2735 . . . . . 6 (π‘š = (π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) β†’ ((π‘š ∘ 𝑉) = 𝐺 ↔ ((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∘ 𝑉) = 𝐺))
2118, 20anbi12d 632 . . . . 5 (π‘š = (π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) β†’ (((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺) ↔ (((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∘ 𝑉) = 𝐺)))
2221rspcev 3613 . . . 4 (((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∈ (𝑃 RingHom 𝑆) ∧ (((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∘ 𝑉) = 𝐺)) β†’ βˆƒπ‘š ∈ (𝑃 RingHom 𝑆)((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺))
23223impb 1116 . . 3 (((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∈ (𝑃 RingHom 𝑆) ∧ ((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∘ 𝐴) = 𝐹 ∧ ((π‘₯ ∈ (Baseβ€˜π‘ƒ) ↦ (𝑆 Ξ£g (𝑦 ∈ {𝑧 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑧 β€œ β„•) ∈ Fin} ↦ ((πΉβ€˜(π‘₯β€˜π‘¦))(.rβ€˜π‘†)((mulGrpβ€˜π‘†) Ξ£g (𝑦 ∘f (.gβ€˜(mulGrpβ€˜π‘†))𝐺)))))) ∘ 𝑉) = 𝐺) β†’ βˆƒπ‘š ∈ (𝑃 RingHom 𝑆)((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺))
2416, 23syl 17 . 2 (πœ‘ β†’ βˆƒπ‘š ∈ (𝑃 RingHom 𝑆)((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺))
25 eqid 2733 . . . . . . . . . 10 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
26 crngring 20068 . . . . . . . . . . 11 (𝑅 ∈ CRing β†’ 𝑅 ∈ Ring)
2711, 26syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝑅 ∈ Ring)
281, 2, 25, 15, 10, 27mplasclf 21626 . . . . . . . . 9 (πœ‘ β†’ 𝐴:(Baseβ€˜π‘…)⟢(Baseβ€˜π‘ƒ))
2928ffund 6722 . . . . . . . 8 (πœ‘ β†’ Fun 𝐴)
30 funcoeqres 6865 . . . . . . . 8 ((Fun 𝐴 ∧ (π‘š ∘ 𝐴) = 𝐹) β†’ (π‘š β†Ύ ran 𝐴) = (𝐹 ∘ ◑𝐴))
3129, 30sylan 581 . . . . . . 7 ((πœ‘ ∧ (π‘š ∘ 𝐴) = 𝐹) β†’ (π‘š β†Ύ ran 𝐴) = (𝐹 ∘ ◑𝐴))
321, 8, 2, 10, 27mvrf2 21552 . . . . . . . . 9 (πœ‘ β†’ 𝑉:𝐼⟢(Baseβ€˜π‘ƒ))
3332ffund 6722 . . . . . . . 8 (πœ‘ β†’ Fun 𝑉)
34 funcoeqres 6865 . . . . . . . 8 ((Fun 𝑉 ∧ (π‘š ∘ 𝑉) = 𝐺) β†’ (π‘š β†Ύ ran 𝑉) = (𝐺 ∘ ◑𝑉))
3533, 34sylan 581 . . . . . . 7 ((πœ‘ ∧ (π‘š ∘ 𝑉) = 𝐺) β†’ (π‘š β†Ύ ran 𝑉) = (𝐺 ∘ ◑𝑉))
3631, 35anim12dan 620 . . . . . 6 ((πœ‘ ∧ ((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺)) β†’ ((π‘š β†Ύ ran 𝐴) = (𝐹 ∘ ◑𝐴) ∧ (π‘š β†Ύ ran 𝑉) = (𝐺 ∘ ◑𝑉)))
3736ex 414 . . . . 5 (πœ‘ β†’ (((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺) β†’ ((π‘š β†Ύ ran 𝐴) = (𝐹 ∘ ◑𝐴) ∧ (π‘š β†Ύ ran 𝑉) = (𝐺 ∘ ◑𝑉))))
38 resundi 5996 . . . . . 6 (π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((π‘š β†Ύ ran 𝐴) βˆͺ (π‘š β†Ύ ran 𝑉))
39 uneq12 4159 . . . . . 6 (((π‘š β†Ύ ran 𝐴) = (𝐹 ∘ ◑𝐴) ∧ (π‘š β†Ύ ran 𝑉) = (𝐺 ∘ ◑𝑉)) β†’ ((π‘š β†Ύ ran 𝐴) βˆͺ (π‘š β†Ύ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉)))
4038, 39eqtrid 2785 . . . . 5 (((π‘š β†Ύ ran 𝐴) = (𝐹 ∘ ◑𝐴) ∧ (π‘š β†Ύ ran 𝑉) = (𝐺 ∘ ◑𝑉)) β†’ (π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉)))
4137, 40syl6 35 . . . 4 (πœ‘ β†’ (((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺) β†’ (π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉))))
4241ralrimivw 3151 . . 3 (πœ‘ β†’ βˆ€π‘š ∈ (𝑃 RingHom 𝑆)(((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺) β†’ (π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉))))
43 eqtr3 2759 . . . . . 6 (((π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉)) ∧ (𝑛 β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉))) β†’ (π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = (𝑛 β†Ύ (ran 𝐴 βˆͺ ran 𝑉)))
44 eqid 2733 . . . . . . . . . . . . 13 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
4544, 10, 11psrassa 21534 . . . . . . . . . . . 12 (πœ‘ β†’ (𝐼 mPwSer 𝑅) ∈ AssAlg)
46 eqid 2733 . . . . . . . . . . . . . 14 (Baseβ€˜(𝐼 mPwSer 𝑅)) = (Baseβ€˜(𝐼 mPwSer 𝑅))
4744, 8, 46, 10, 27mvrf 21544 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑉:𝐼⟢(Baseβ€˜(𝐼 mPwSer 𝑅)))
4847frnd 6726 . . . . . . . . . . . 12 (πœ‘ β†’ ran 𝑉 βŠ† (Baseβ€˜(𝐼 mPwSer 𝑅)))
49 eqid 2733 . . . . . . . . . . . . 13 (AlgSpanβ€˜(𝐼 mPwSer 𝑅)) = (AlgSpanβ€˜(𝐼 mPwSer 𝑅))
50 eqid 2733 . . . . . . . . . . . . 13 (algScβ€˜(𝐼 mPwSer 𝑅)) = (algScβ€˜(𝐼 mPwSer 𝑅))
51 eqid 2733 . . . . . . . . . . . . 13 (mrClsβ€˜(SubRingβ€˜(𝐼 mPwSer 𝑅))) = (mrClsβ€˜(SubRingβ€˜(𝐼 mPwSer 𝑅)))
5249, 50, 51, 46aspval2 21452 . . . . . . . . . . . 12 (((𝐼 mPwSer 𝑅) ∈ AssAlg ∧ ran 𝑉 βŠ† (Baseβ€˜(𝐼 mPwSer 𝑅))) β†’ ((AlgSpanβ€˜(𝐼 mPwSer 𝑅))β€˜ran 𝑉) = ((mrClsβ€˜(SubRingβ€˜(𝐼 mPwSer 𝑅)))β€˜(ran (algScβ€˜(𝐼 mPwSer 𝑅)) βˆͺ ran 𝑉)))
5345, 48, 52syl2anc 585 . . . . . . . . . . 11 (πœ‘ β†’ ((AlgSpanβ€˜(𝐼 mPwSer 𝑅))β€˜ran 𝑉) = ((mrClsβ€˜(SubRingβ€˜(𝐼 mPwSer 𝑅)))β€˜(ran (algScβ€˜(𝐼 mPwSer 𝑅)) βˆͺ ran 𝑉)))
541, 44, 8, 49, 10, 11mplbas2 21597 . . . . . . . . . . 11 (πœ‘ β†’ ((AlgSpanβ€˜(𝐼 mPwSer 𝑅))β€˜ran 𝑉) = (Baseβ€˜π‘ƒ))
5544, 1, 2, 10, 27mplsubrg 21564 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (Baseβ€˜π‘ƒ) ∈ (SubRingβ€˜(𝐼 mPwSer 𝑅)))
561, 44, 2mplval2 21555 . . . . . . . . . . . . . . . 16 𝑃 = ((𝐼 mPwSer 𝑅) β†Ύs (Baseβ€˜π‘ƒ))
5756subsubrg2 20346 . . . . . . . . . . . . . . 15 ((Baseβ€˜π‘ƒ) ∈ (SubRingβ€˜(𝐼 mPwSer 𝑅)) β†’ (SubRingβ€˜π‘ƒ) = ((SubRingβ€˜(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Baseβ€˜π‘ƒ)))
5855, 57syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ (SubRingβ€˜π‘ƒ) = ((SubRingβ€˜(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Baseβ€˜π‘ƒ)))
5958fveq2d 6896 . . . . . . . . . . . . 13 (πœ‘ β†’ (mrClsβ€˜(SubRingβ€˜π‘ƒ)) = (mrClsβ€˜((SubRingβ€˜(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Baseβ€˜π‘ƒ))))
6050, 56ressascl 21450 . . . . . . . . . . . . . . . . 17 ((Baseβ€˜π‘ƒ) ∈ (SubRingβ€˜(𝐼 mPwSer 𝑅)) β†’ (algScβ€˜(𝐼 mPwSer 𝑅)) = (algScβ€˜π‘ƒ))
6155, 60syl 17 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (algScβ€˜(𝐼 mPwSer 𝑅)) = (algScβ€˜π‘ƒ))
6215, 61eqtr4id 2792 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐴 = (algScβ€˜(𝐼 mPwSer 𝑅)))
6362rneqd 5938 . . . . . . . . . . . . . 14 (πœ‘ β†’ ran 𝐴 = ran (algScβ€˜(𝐼 mPwSer 𝑅)))
6463uneq1d 4163 . . . . . . . . . . . . 13 (πœ‘ β†’ (ran 𝐴 βˆͺ ran 𝑉) = (ran (algScβ€˜(𝐼 mPwSer 𝑅)) βˆͺ ran 𝑉))
6559, 64fveq12d 6899 . . . . . . . . . . . 12 (πœ‘ β†’ ((mrClsβ€˜(SubRingβ€˜π‘ƒ))β€˜(ran 𝐴 βˆͺ ran 𝑉)) = ((mrClsβ€˜((SubRingβ€˜(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Baseβ€˜π‘ƒ)))β€˜(ran (algScβ€˜(𝐼 mPwSer 𝑅)) βˆͺ ran 𝑉)))
66 assaring 21416 . . . . . . . . . . . . . 14 ((𝐼 mPwSer 𝑅) ∈ AssAlg β†’ (𝐼 mPwSer 𝑅) ∈ Ring)
6746subrgmre 20344 . . . . . . . . . . . . . 14 ((𝐼 mPwSer 𝑅) ∈ Ring β†’ (SubRingβ€˜(𝐼 mPwSer 𝑅)) ∈ (Mooreβ€˜(Baseβ€˜(𝐼 mPwSer 𝑅))))
6845, 66, 673syl 18 . . . . . . . . . . . . 13 (πœ‘ β†’ (SubRingβ€˜(𝐼 mPwSer 𝑅)) ∈ (Mooreβ€˜(Baseβ€˜(𝐼 mPwSer 𝑅))))
6928frnd 6726 . . . . . . . . . . . . . . 15 (πœ‘ β†’ ran 𝐴 βŠ† (Baseβ€˜π‘ƒ))
7063, 69eqsstrrd 4022 . . . . . . . . . . . . . 14 (πœ‘ β†’ ran (algScβ€˜(𝐼 mPwSer 𝑅)) βŠ† (Baseβ€˜π‘ƒ))
7132frnd 6726 . . . . . . . . . . . . . 14 (πœ‘ β†’ ran 𝑉 βŠ† (Baseβ€˜π‘ƒ))
7270, 71unssd 4187 . . . . . . . . . . . . 13 (πœ‘ β†’ (ran (algScβ€˜(𝐼 mPwSer 𝑅)) βˆͺ ran 𝑉) βŠ† (Baseβ€˜π‘ƒ))
73 eqid 2733 . . . . . . . . . . . . . 14 (mrClsβ€˜((SubRingβ€˜(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Baseβ€˜π‘ƒ))) = (mrClsβ€˜((SubRingβ€˜(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Baseβ€˜π‘ƒ)))
7451, 73submrc 17572 . . . . . . . . . . . . 13 (((SubRingβ€˜(𝐼 mPwSer 𝑅)) ∈ (Mooreβ€˜(Baseβ€˜(𝐼 mPwSer 𝑅))) ∧ (Baseβ€˜π‘ƒ) ∈ (SubRingβ€˜(𝐼 mPwSer 𝑅)) ∧ (ran (algScβ€˜(𝐼 mPwSer 𝑅)) βˆͺ ran 𝑉) βŠ† (Baseβ€˜π‘ƒ)) β†’ ((mrClsβ€˜((SubRingβ€˜(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Baseβ€˜π‘ƒ)))β€˜(ran (algScβ€˜(𝐼 mPwSer 𝑅)) βˆͺ ran 𝑉)) = ((mrClsβ€˜(SubRingβ€˜(𝐼 mPwSer 𝑅)))β€˜(ran (algScβ€˜(𝐼 mPwSer 𝑅)) βˆͺ ran 𝑉)))
7568, 55, 72, 74syl3anc 1372 . . . . . . . . . . . 12 (πœ‘ β†’ ((mrClsβ€˜((SubRingβ€˜(𝐼 mPwSer 𝑅)) ∩ 𝒫 (Baseβ€˜π‘ƒ)))β€˜(ran (algScβ€˜(𝐼 mPwSer 𝑅)) βˆͺ ran 𝑉)) = ((mrClsβ€˜(SubRingβ€˜(𝐼 mPwSer 𝑅)))β€˜(ran (algScβ€˜(𝐼 mPwSer 𝑅)) βˆͺ ran 𝑉)))
7665, 75eqtr2d 2774 . . . . . . . . . . 11 (πœ‘ β†’ ((mrClsβ€˜(SubRingβ€˜(𝐼 mPwSer 𝑅)))β€˜(ran (algScβ€˜(𝐼 mPwSer 𝑅)) βˆͺ ran 𝑉)) = ((mrClsβ€˜(SubRingβ€˜π‘ƒ))β€˜(ran 𝐴 βˆͺ ran 𝑉)))
7753, 54, 763eqtr3d 2781 . . . . . . . . . 10 (πœ‘ β†’ (Baseβ€˜π‘ƒ) = ((mrClsβ€˜(SubRingβ€˜π‘ƒ))β€˜(ran 𝐴 βˆͺ ran 𝑉)))
7877ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 βˆͺ ran 𝑉) βŠ† dom (π‘š ∩ 𝑛)) β†’ (Baseβ€˜π‘ƒ) = ((mrClsβ€˜(SubRingβ€˜π‘ƒ))β€˜(ran 𝐴 βˆͺ ran 𝑉)))
791mplring 21578 . . . . . . . . . . . . 13 ((𝐼 ∈ π‘Š ∧ 𝑅 ∈ Ring) β†’ 𝑃 ∈ Ring)
8010, 27, 79syl2anc 585 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑃 ∈ Ring)
812subrgmre 20344 . . . . . . . . . . . 12 (𝑃 ∈ Ring β†’ (SubRingβ€˜π‘ƒ) ∈ (Mooreβ€˜(Baseβ€˜π‘ƒ)))
8280, 81syl 17 . . . . . . . . . . 11 (πœ‘ β†’ (SubRingβ€˜π‘ƒ) ∈ (Mooreβ€˜(Baseβ€˜π‘ƒ)))
8382ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 βˆͺ ran 𝑉) βŠ† dom (π‘š ∩ 𝑛)) β†’ (SubRingβ€˜π‘ƒ) ∈ (Mooreβ€˜(Baseβ€˜π‘ƒ)))
84 simpr 486 . . . . . . . . . 10 (((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 βˆͺ ran 𝑉) βŠ† dom (π‘š ∩ 𝑛)) β†’ (ran 𝐴 βˆͺ ran 𝑉) βŠ† dom (π‘š ∩ 𝑛))
85 rhmeql 20350 . . . . . . . . . . 11 ((π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆)) β†’ dom (π‘š ∩ 𝑛) ∈ (SubRingβ€˜π‘ƒ))
8685ad2antlr 726 . . . . . . . . . 10 (((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 βˆͺ ran 𝑉) βŠ† dom (π‘š ∩ 𝑛)) β†’ dom (π‘š ∩ 𝑛) ∈ (SubRingβ€˜π‘ƒ))
87 eqid 2733 . . . . . . . . . . 11 (mrClsβ€˜(SubRingβ€˜π‘ƒ)) = (mrClsβ€˜(SubRingβ€˜π‘ƒ))
8887mrcsscl 17564 . . . . . . . . . 10 (((SubRingβ€˜π‘ƒ) ∈ (Mooreβ€˜(Baseβ€˜π‘ƒ)) ∧ (ran 𝐴 βˆͺ ran 𝑉) βŠ† dom (π‘š ∩ 𝑛) ∧ dom (π‘š ∩ 𝑛) ∈ (SubRingβ€˜π‘ƒ)) β†’ ((mrClsβ€˜(SubRingβ€˜π‘ƒ))β€˜(ran 𝐴 βˆͺ ran 𝑉)) βŠ† dom (π‘š ∩ 𝑛))
8983, 84, 86, 88syl3anc 1372 . . . . . . . . 9 (((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 βˆͺ ran 𝑉) βŠ† dom (π‘š ∩ 𝑛)) β†’ ((mrClsβ€˜(SubRingβ€˜π‘ƒ))β€˜(ran 𝐴 βˆͺ ran 𝑉)) βŠ† dom (π‘š ∩ 𝑛))
9078, 89eqsstrd 4021 . . . . . . . 8 (((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) ∧ (ran 𝐴 βˆͺ ran 𝑉) βŠ† dom (π‘š ∩ 𝑛)) β†’ (Baseβ€˜π‘ƒ) βŠ† dom (π‘š ∩ 𝑛))
9190ex 414 . . . . . . 7 ((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) β†’ ((ran 𝐴 βˆͺ ran 𝑉) βŠ† dom (π‘š ∩ 𝑛) β†’ (Baseβ€˜π‘ƒ) βŠ† dom (π‘š ∩ 𝑛)))
92 simprl 770 . . . . . . . . 9 ((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) β†’ π‘š ∈ (𝑃 RingHom 𝑆))
932, 3rhmf 20263 . . . . . . . . 9 (π‘š ∈ (𝑃 RingHom 𝑆) β†’ π‘š:(Baseβ€˜π‘ƒ)⟢𝐢)
94 ffn 6718 . . . . . . . . 9 (π‘š:(Baseβ€˜π‘ƒ)⟢𝐢 β†’ π‘š Fn (Baseβ€˜π‘ƒ))
9592, 93, 943syl 18 . . . . . . . 8 ((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) β†’ π‘š Fn (Baseβ€˜π‘ƒ))
96 simprr 772 . . . . . . . . 9 ((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) β†’ 𝑛 ∈ (𝑃 RingHom 𝑆))
972, 3rhmf 20263 . . . . . . . . 9 (𝑛 ∈ (𝑃 RingHom 𝑆) β†’ 𝑛:(Baseβ€˜π‘ƒ)⟢𝐢)
98 ffn 6718 . . . . . . . . 9 (𝑛:(Baseβ€˜π‘ƒ)⟢𝐢 β†’ 𝑛 Fn (Baseβ€˜π‘ƒ))
9996, 97, 983syl 18 . . . . . . . 8 ((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) β†’ 𝑛 Fn (Baseβ€˜π‘ƒ))
10069adantr 482 . . . . . . . . 9 ((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) β†’ ran 𝐴 βŠ† (Baseβ€˜π‘ƒ))
10171adantr 482 . . . . . . . . 9 ((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) β†’ ran 𝑉 βŠ† (Baseβ€˜π‘ƒ))
102100, 101unssd 4187 . . . . . . . 8 ((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) β†’ (ran 𝐴 βˆͺ ran 𝑉) βŠ† (Baseβ€˜π‘ƒ))
103 fnreseql 7050 . . . . . . . 8 ((π‘š Fn (Baseβ€˜π‘ƒ) ∧ 𝑛 Fn (Baseβ€˜π‘ƒ) ∧ (ran 𝐴 βˆͺ ran 𝑉) βŠ† (Baseβ€˜π‘ƒ)) β†’ ((π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = (𝑛 β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) ↔ (ran 𝐴 βˆͺ ran 𝑉) βŠ† dom (π‘š ∩ 𝑛)))
10495, 99, 102, 103syl3anc 1372 . . . . . . 7 ((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) β†’ ((π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = (𝑛 β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) ↔ (ran 𝐴 βˆͺ ran 𝑉) βŠ† dom (π‘š ∩ 𝑛)))
105 fneqeql2 7049 . . . . . . . 8 ((π‘š Fn (Baseβ€˜π‘ƒ) ∧ 𝑛 Fn (Baseβ€˜π‘ƒ)) β†’ (π‘š = 𝑛 ↔ (Baseβ€˜π‘ƒ) βŠ† dom (π‘š ∩ 𝑛)))
10695, 99, 105syl2anc 585 . . . . . . 7 ((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) β†’ (π‘š = 𝑛 ↔ (Baseβ€˜π‘ƒ) βŠ† dom (π‘š ∩ 𝑛)))
10791, 104, 1063imtr4d 294 . . . . . 6 ((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) β†’ ((π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = (𝑛 β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) β†’ π‘š = 𝑛))
10843, 107syl5 34 . . . . 5 ((πœ‘ ∧ (π‘š ∈ (𝑃 RingHom 𝑆) ∧ 𝑛 ∈ (𝑃 RingHom 𝑆))) β†’ (((π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉)) ∧ (𝑛 β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉))) β†’ π‘š = 𝑛))
109108ralrimivva 3201 . . . 4 (πœ‘ β†’ βˆ€π‘š ∈ (𝑃 RingHom 𝑆)βˆ€π‘› ∈ (𝑃 RingHom 𝑆)(((π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉)) ∧ (𝑛 β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉))) β†’ π‘š = 𝑛))
110 reseq1 5976 . . . . . 6 (π‘š = 𝑛 β†’ (π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = (𝑛 β†Ύ (ran 𝐴 βˆͺ ran 𝑉)))
111110eqeq1d 2735 . . . . 5 (π‘š = 𝑛 β†’ ((π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉)) ↔ (𝑛 β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉))))
112111rmo4 3727 . . . 4 (βˆƒ*π‘š ∈ (𝑃 RingHom 𝑆)(π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉)) ↔ βˆ€π‘š ∈ (𝑃 RingHom 𝑆)βˆ€π‘› ∈ (𝑃 RingHom 𝑆)(((π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉)) ∧ (𝑛 β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉))) β†’ π‘š = 𝑛))
113109, 112sylibr 233 . . 3 (πœ‘ β†’ βˆƒ*π‘š ∈ (𝑃 RingHom 𝑆)(π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉)))
114 rmoim 3737 . . 3 (βˆ€π‘š ∈ (𝑃 RingHom 𝑆)(((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺) β†’ (π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉))) β†’ (βˆƒ*π‘š ∈ (𝑃 RingHom 𝑆)(π‘š β†Ύ (ran 𝐴 βˆͺ ran 𝑉)) = ((𝐹 ∘ ◑𝐴) βˆͺ (𝐺 ∘ ◑𝑉)) β†’ βˆƒ*π‘š ∈ (𝑃 RingHom 𝑆)((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺)))
11542, 113, 114sylc 65 . 2 (πœ‘ β†’ βˆƒ*π‘š ∈ (𝑃 RingHom 𝑆)((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺))
116 reu5 3379 . 2 (βˆƒ!π‘š ∈ (𝑃 RingHom 𝑆)((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺) ↔ (βˆƒπ‘š ∈ (𝑃 RingHom 𝑆)((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺) ∧ βˆƒ*π‘š ∈ (𝑃 RingHom 𝑆)((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺)))
11724, 115, 116sylanbrc 584 1 (πœ‘ β†’ βˆƒ!π‘š ∈ (𝑃 RingHom 𝑆)((π‘š ∘ 𝐴) = 𝐹 ∧ (π‘š ∘ 𝑉) = 𝐺))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  βˆƒ!wreu 3375  βˆƒ*wrmo 3376  {crab 3433   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4603   ↦ cmpt 5232  β—‘ccnv 5676  dom cdm 5677  ran crn 5678   β†Ύ cres 5679   β€œ cima 5680   ∘ ccom 5681  Fun wfun 6538   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∘f cof 7668   ↑m cmap 8820  Fincfn 8939  β„•cn 12212  β„•0cn0 12472  Basecbs 17144  .rcmulr 17198   Ξ£g cgsu 17386  Moorecmre 17526  mrClscmrc 17527  .gcmg 18950  mulGrpcmgp 19987  Ringcrg 20056  CRingccrg 20057   RingHom crh 20248  SubRingcsubrg 20315  AssAlgcasa 21405  AlgSpancasp 21406  algSccascl 21407   mPwSer cmps 21457   mVar cmvr 21458   mPoly cmpl 21459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-ofr 7671  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9362  df-sup 9437  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-fzo 13628  df-seq 13967  df-hash 14291  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-hom 17221  df-cco 17222  df-0g 17387  df-gsum 17388  df-prds 17393  df-pws 17395  df-mre 17530  df-mrc 17531  df-acs 17533  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-submnd 18672  df-grp 18822  df-minusg 18823  df-sbg 18824  df-mulg 18951  df-subg 19003  df-ghm 19090  df-cntz 19181  df-cmn 19650  df-abl 19651  df-mgp 19988  df-ur 20005  df-srg 20010  df-ring 20058  df-cring 20059  df-rnghom 20251  df-subrg 20317  df-lmod 20473  df-lss 20543  df-lsp 20583  df-assa 21408  df-asp 21409  df-ascl 21410  df-psr 21462  df-mvr 21463  df-mpl 21464
This theorem is referenced by:  evlsval2  21650  evlsval3  41131
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