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Theorem chcoeffeqlem 20509
 Description: Lemma for chcoeffeq 20510. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) (Revised by AV, 15-Dec-2019.)
Hypotheses
Ref Expression
chcoeffeq.a 𝐴 = (𝑁 Mat 𝑅)
chcoeffeq.b 𝐵 = (Base‘𝐴)
chcoeffeq.p 𝑃 = (Poly1𝑅)
chcoeffeq.y 𝑌 = (𝑁 Mat 𝑃)
chcoeffeq.r × = (.r𝑌)
chcoeffeq.s = (-g𝑌)
chcoeffeq.0 0 = (0g𝑌)
chcoeffeq.t 𝑇 = (𝑁 matToPolyMat 𝑅)
chcoeffeq.c 𝐶 = (𝑁 CharPlyMat 𝑅)
chcoeffeq.k 𝐾 = (𝐶𝑀)
chcoeffeq.g 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))
chcoeffeq.w 𝑊 = (Base‘𝑌)
chcoeffeq.1 1 = (1r𝐴)
chcoeffeq.m = ( ·𝑠𝐴)
chcoeffeq.u 𝑈 = (𝑁 cPolyMatToMat 𝑅)
Assertion
Ref Expression
chcoeffeqlem (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 )))
Distinct variable groups:   𝐴,𝑛   𝐵,𝑛   𝑛,𝐺   𝑛,𝐾   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑈,𝑛   𝑛,𝑌   1 ,𝑛   ,𝑛   𝑛,𝑏   𝑛,𝑠
Allowed substitution hints:   𝐴(𝑠,𝑏)   𝐵(𝑠,𝑏)   𝐶(𝑛,𝑠,𝑏)   𝑃(𝑛,𝑠,𝑏)   𝑅(𝑠,𝑏)   𝑇(𝑛,𝑠,𝑏)   × (𝑛,𝑠,𝑏)   𝑈(𝑠,𝑏)   1 (𝑠,𝑏)   𝐺(𝑠,𝑏)   (𝑠,𝑏)   𝐾(𝑠,𝑏)   𝑀(𝑠,𝑏)   (𝑛,𝑠,𝑏)   𝑁(𝑠,𝑏)   𝑊(𝑛,𝑠,𝑏)   𝑌(𝑠,𝑏)   0 (𝑛,𝑠,𝑏)

Proof of Theorem chcoeffeqlem
Dummy variable 𝑙 is distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . 5 (Poly1𝐴) = (Poly1𝐴)
2 eqid 2610 . . . . 5 (var1𝐴) = (var1𝐴)
3 eqid 2610 . . . . 5 (.g‘(mulGrp‘(Poly1𝐴))) = (.g‘(mulGrp‘(Poly1𝐴)))
4 crngring 18381 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5 chcoeffeq.a . . . . . . . . 9 𝐴 = (𝑁 Mat 𝑅)
65matring 20068 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
74, 6sylan2 490 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
873adant3 1074 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ Ring)
98adantr 480 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐴 ∈ Ring)
10 chcoeffeq.b . . . . 5 𝐵 = (Base‘𝐴)
11 eqid 2610 . . . . 5 ( ·𝑠 ‘(Poly1𝐴)) = ( ·𝑠 ‘(Poly1𝐴))
12 eqid 2610 . . . . 5 (0g𝐴) = (0g𝐴)
13 chcoeffeq.p . . . . . . . 8 𝑃 = (Poly1𝑅)
14 chcoeffeq.y . . . . . . . 8 𝑌 = (𝑁 Mat 𝑃)
15 chcoeffeq.t . . . . . . . 8 𝑇 = (𝑁 matToPolyMat 𝑅)
16 chcoeffeq.r . . . . . . . 8 × = (.r𝑌)
17 chcoeffeq.s . . . . . . . 8 = (-g𝑌)
18 chcoeffeq.0 . . . . . . . 8 0 = (0g𝑌)
19 chcoeffeq.g . . . . . . . 8 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))
20 eqid 2610 . . . . . . . 8 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
21 eqid 2610 . . . . . . . 8 ( ·𝑠𝑌) = ( ·𝑠𝑌)
22 eqid 2610 . . . . . . . 8 (1r𝑌) = (1r𝑌)
23 eqid 2610 . . . . . . . 8 (var1𝑅) = (var1𝑅)
24 eqid 2610 . . . . . . . 8 (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) = (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))
25 eqid 2610 . . . . . . . 8 (𝑁 maAdju 𝑃) = (𝑁 maAdju 𝑃)
26 chcoeffeq.w . . . . . . . 8 𝑊 = (Base‘𝑌)
27 chcoeffeq.u . . . . . . . 8 𝑈 = (𝑁 cPolyMatToMat 𝑅)
285, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 1, 2, 11, 3, 27cpmadumatpolylem1 20505 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝑈𝐺) ∈ (𝐵𝑚0))
2928anasss 677 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑈𝐺) ∈ (𝐵𝑚0))
305, 10, 13, 14, 16, 17, 18, 15, 19, 20chfacfisfcpmat 20479 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
314, 30syl3anl2 1367 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
3231adantr 480 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
33 fvco3 6185 . . . . . . . . . 10 ((𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ0) → ((𝑈𝐺)‘𝑙) = (𝑈‘(𝐺𝑙)))
3433eqcomd 2616 . . . . . . . . 9 ((𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺𝑙)) = ((𝑈𝐺)‘𝑙))
3532, 34sylan 487 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺𝑙)) = ((𝑈𝐺)‘𝑙))
36 elmapi 7765 . . . . . . . . . 10 ((𝑈𝐺) ∈ (𝐵𝑚0) → (𝑈𝐺):ℕ0𝐵)
3736adantl 481 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) → (𝑈𝐺):ℕ0𝐵)
3837ffvelrnda 6267 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) ∧ 𝑙 ∈ ℕ0) → ((𝑈𝐺)‘𝑙) ∈ 𝐵)
3935, 38eqeltrd 2688 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺𝑙)) ∈ 𝐵)
4039ralrimiva 2949 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) → ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺𝑙)) ∈ 𝐵)
4129, 40mpdan 699 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺𝑙)) ∈ 𝐵)
424anim2i 591 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
43423adant3 1074 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
4443adantr 480 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
455, 10, 20, 27cpm2mf 20376 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
4644, 45syl 17 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
47 fcompt 6306 . . . . . . 7 ((𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)) → (𝑈𝐺) = (𝑙 ∈ ℕ0 ↦ (𝑈‘(𝐺𝑙))))
4846, 31, 47syl2anc 691 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑈𝐺) = (𝑙 ∈ ℕ0 ↦ (𝑈‘(𝐺𝑙))))
495, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 1, 2, 11, 3, 27cpmadumatpolylem2 20506 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝑈𝐺) finSupp (0g𝐴))
5049anasss 677 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑈𝐺) finSupp (0g𝐴))
5148, 50eqbrtrrd 4607 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑙 ∈ ℕ0 ↦ (𝑈‘(𝐺𝑙))) finSupp (0g𝐴))
52 simpll1 1093 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → 𝑁 ∈ Fin)
5343ad2ant2 1076 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
5453ad2antrr 758 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → 𝑅 ∈ Ring)
55 chcoeffeq.k . . . . . . . . . 10 𝐾 = (𝐶𝑀)
56 chcoeffeq.c . . . . . . . . . . 11 𝐶 = (𝑁 CharPlyMat 𝑅)
57 eqid 2610 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘𝑃)
5856, 5, 10, 13, 57chpmatply1 20456 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐶𝑀) ∈ (Base‘𝑃))
5955, 58syl5eqel 2692 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐾 ∈ (Base‘𝑃))
60 eqid 2610 . . . . . . . . . 10 (coe1𝐾) = (coe1𝐾)
61 eqid 2610 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
6260, 57, 13, 61coe1fvalcl 19403 . . . . . . . . 9 ((𝐾 ∈ (Base‘𝑃) ∧ 𝑙 ∈ ℕ0) → ((coe1𝐾)‘𝑙) ∈ (Base‘𝑅))
6359, 62sylan 487 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑙 ∈ ℕ0) → ((coe1𝐾)‘𝑙) ∈ (Base‘𝑅))
6463adantlr 747 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → ((coe1𝐾)‘𝑙) ∈ (Base‘𝑅))
65 chcoeffeq.1 . . . . . . . . . 10 1 = (1r𝐴)
6610, 65ringidcl 18391 . . . . . . . . 9 (𝐴 ∈ Ring → 1𝐵)
678, 66syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 1𝐵)
6867ad2antrr 758 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → 1𝐵)
69 chcoeffeq.m . . . . . . . 8 = ( ·𝑠𝐴)
7061, 5, 10, 69matvscl 20056 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (((coe1𝐾)‘𝑙) ∈ (Base‘𝑅) ∧ 1𝐵)) → (((coe1𝐾)‘𝑙) 1 ) ∈ 𝐵)
7152, 54, 64, 68, 70syl22anc 1319 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → (((coe1𝐾)‘𝑙) 1 ) ∈ 𝐵)
7271ralrimiva 2949 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ∀𝑙 ∈ ℕ0 (((coe1𝐾)‘𝑙) 1 ) ∈ 𝐵)
73 nn0ex 11175 . . . . . . 7 0 ∈ V
7473a1i 11 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ℕ0 ∈ V)
755matlmod 20054 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)
764, 75sylan2 490 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ LMod)
77763adant3 1074 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ LMod)
7877adantr 480 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐴 ∈ LMod)
79 eqidd 2611 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (Scalar‘𝐴) = (Scalar‘𝐴))
80 fvex 6113 . . . . . . 7 ((coe1𝐾)‘𝑙) ∈ V
8180a1i 11 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → ((coe1𝐾)‘𝑙) ∈ V)
82 eqid 2610 . . . . . 6 (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴))
835matsca2 20045 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝐴))
84833adant3 1074 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 = (Scalar‘𝐴))
8584, 53eqeltrrd 2689 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Scalar‘𝐴) ∈ Ring)
8684eqcomd 2616 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Scalar‘𝐴) = 𝑅)
8786fveq2d 6107 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Poly1‘(Scalar‘𝐴)) = (Poly1𝑅))
8887, 13syl6eqr 2662 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Poly1‘(Scalar‘𝐴)) = 𝑃)
8988fveq2d 6107 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Base‘(Poly1‘(Scalar‘𝐴))) = (Base‘𝑃))
9059, 89eleqtrrd 2691 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐾 ∈ (Base‘(Poly1‘(Scalar‘𝐴))))
91 eqid 2610 . . . . . . . . 9 (Poly1‘(Scalar‘𝐴)) = (Poly1‘(Scalar‘𝐴))
92 eqid 2610 . . . . . . . . 9 (Base‘(Poly1‘(Scalar‘𝐴))) = (Base‘(Poly1‘(Scalar‘𝐴)))
9391, 92, 82mptcoe1fsupp 19406 . . . . . . . 8 (((Scalar‘𝐴) ∈ Ring ∧ 𝐾 ∈ (Base‘(Poly1‘(Scalar‘𝐴)))) → (𝑙 ∈ ℕ0 ↦ ((coe1𝐾)‘𝑙)) finSupp (0g‘(Scalar‘𝐴)))
9485, 90, 93syl2anc 691 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑙 ∈ ℕ0 ↦ ((coe1𝐾)‘𝑙)) finSupp (0g‘(Scalar‘𝐴)))
9594adantr 480 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑙 ∈ ℕ0 ↦ ((coe1𝐾)‘𝑙)) finSupp (0g‘(Scalar‘𝐴)))
9674, 78, 79, 10, 81, 68, 12, 82, 69, 95mptscmfsupp0 18751 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑙 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑙) 1 )) finSupp (0g𝐴))
97 fveq2 6103 . . . . . . . . . 10 (𝑛 = 𝑙 → (𝐺𝑛) = (𝐺𝑙))
9897fveq2d 6107 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑈‘(𝐺𝑛)) = (𝑈‘(𝐺𝑙)))
99 oveq1 6556 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)) = (𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))
10098, 99oveq12d 6567 . . . . . . . 8 (𝑛 = 𝑙 → ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))) = ((𝑈‘(𝐺𝑙))( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
101100cbvmptv 4678 . . . . . . 7 (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))) = (𝑙 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑙))( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
102101oveq2i 6560 . . . . . 6 ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑙 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑙))( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))
103102a1i 11 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑙 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑙))( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))))
104 fveq2 6103 . . . . . . . . . 10 (𝑛 = 𝑙 → ((coe1𝐾)‘𝑛) = ((coe1𝐾)‘𝑙))
105104oveq1d 6564 . . . . . . . . 9 (𝑛 = 𝑙 → (((coe1𝐾)‘𝑛) 1 ) = (((coe1𝐾)‘𝑙) 1 ))
106105, 99oveq12d 6567 . . . . . . . 8 (𝑛 = 𝑙 → ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))) = ((((coe1𝐾)‘𝑙) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
107106cbvmptv 4678 . . . . . . 7 (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))) = (𝑙 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑙) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
108107oveq2i 6560 . . . . . 6 ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑙 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑙) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))
109108a1i 11 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑙 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑙) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))))
1101, 2, 3, 9, 10, 11, 12, 41, 51, 72, 96, 103, 109gsumply1eq 19496 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) ↔ ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺𝑙)) = (((coe1𝐾)‘𝑙) 1 )))
111110biimpa 500 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))) → ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺𝑙)) = (((coe1𝐾)‘𝑙) 1 ))
11298, 105eqeq12d 2625 . . . 4 (𝑛 = 𝑙 → ((𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ) ↔ (𝑈‘(𝐺𝑙)) = (((coe1𝐾)‘𝑙) 1 )))
113112cbvralv 3147 . . 3 (∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ) ↔ ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺𝑙)) = (((coe1𝐾)‘𝑙) 1 ))
114111, 113sylibr 223 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ))
115114ex 449 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Fincfn 7841   finSupp cfsupp 8158  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ...cfz 12197  Basecbs 15695  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923   Σg cgsu 15924  -gcsg 17247  .gcmg 17363  mulGrpcmgp 18312  1rcur 18324  Ringcrg 18370  CRingccrg 18371  LModclmod 18686  var1cv1 19367  Poly1cpl1 19368  coe1cco1 19369   Mat cmat 20032   maAdju cmadu 20257   ConstPolyMat ccpmat 20327   matToPolyMat cmat2pmat 20328   cPolyMatToMat ccpmat2mat 20329   CharPlyMat cchpmat 20450 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  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-xor 1457  df-tru 1478  df-fal 1481  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-ot 4134  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-tpos 7239  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-sup 8231  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-div 10564  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-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-reverse 13160  df-s2 13444  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-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  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-gim 17524  df-cntz 17573  df-oppg 17599  df-symg 17621  df-pmtr 17685  df-psgn 17734  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-srg 18329  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-rnghom 18538  df-drng 18572  df-subrg 18601  df-lmod 18688  df-lss 18754  df-sra 18993  df-rgmod 18994  df-ascl 19135  df-psr 19177  df-mvr 19178  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-vr1 19372  df-ply1 19373  df-coe1 19374  df-cnfld 19568  df-zring 19638  df-zrh 19671  df-dsmm 19895  df-frlm 19910  df-mamu 20009  df-mat 20033  df-mdet 20210  df-cpmat 20330  df-mat2pmat 20331  df-cpmat2mat 20332  df-chpmat 20451 This theorem is referenced by:  chcoeffeq  20510
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