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Theorem chcoeffeqlem 22045
Description: Lemma for chcoeffeq 22046. (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 2740 . . . . 5 (Poly1𝐴) = (Poly1𝐴)
2 eqid 2740 . . . . 5 (var1𝐴) = (var1𝐴)
3 eqid 2740 . . . . 5 (.g‘(mulGrp‘(Poly1𝐴))) = (.g‘(mulGrp‘(Poly1𝐴)))
4 crngring 19806 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5 chcoeffeq.a . . . . . . . . 9 𝐴 = (𝑁 Mat 𝑅)
65matring 21603 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
74, 6sylan2 593 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
873adant3 1131 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ Ring)
98adantr 481 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝐴 ∈ Ring)
10 chcoeffeq.b . . . . 5 𝐵 = (Base‘𝐴)
11 eqid 2740 . . . . 5 ( ·𝑠 ‘(Poly1𝐴)) = ( ·𝑠 ‘(Poly1𝐴))
12 eqid 2740 . . . . 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 2740 . . . . . . . 8 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
21 eqid 2740 . . . . . . . 8 ( ·𝑠𝑌) = ( ·𝑠𝑌)
22 eqid 2740 . . . . . . . 8 (1r𝑌) = (1r𝑌)
23 eqid 2740 . . . . . . . 8 (var1𝑅) = (var1𝑅)
24 eqid 2740 . . . . . . . 8 (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) = (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))
25 eqid 2740 . . . . . . . 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 22041 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝑈𝐺) ∈ (𝐵m0))
2928anasss 467 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑈𝐺) ∈ (𝐵m0))
305, 10, 13, 14, 16, 17, 18, 15, 19, 20chfacfisfcpmat 22015 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
314, 30syl3anl2 1412 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
3231adantr 481 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵m0)) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
33 fvco3 6864 . . . . . . . . . 10 ((𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ0) → ((𝑈𝐺)‘𝑙) = (𝑈‘(𝐺𝑙)))
3433eqcomd 2746 . . . . . . . . 9 ((𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺𝑙)) = ((𝑈𝐺)‘𝑙))
3532, 34sylan 580 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵m0)) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺𝑙)) = ((𝑈𝐺)‘𝑙))
36 elmapi 8629 . . . . . . . . . 10 ((𝑈𝐺) ∈ (𝐵m0) → (𝑈𝐺):ℕ0𝐵)
3736adantl 482 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵m0)) → (𝑈𝐺):ℕ0𝐵)
3837ffvelrnda 6958 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵m0)) ∧ 𝑙 ∈ ℕ0) → ((𝑈𝐺)‘𝑙) ∈ 𝐵)
3935, 38eqeltrd 2841 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵m0)) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺𝑙)) ∈ 𝐵)
4039ralrimiva 3110 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵m0)) → ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺𝑙)) ∈ 𝐵)
4129, 40mpdan 684 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺𝑙)) ∈ 𝐵)
424anim2i 617 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
43423adant3 1131 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
4443adantr 481 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
455, 10, 20, 27cpm2mf 21912 . . . . . . . 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 7002 . . . . . . 7 ((𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)) → (𝑈𝐺) = (𝑙 ∈ ℕ0 ↦ (𝑈‘(𝐺𝑙))))
4846, 31, 47syl2anc 584 . . . . . 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 22042 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝑈𝐺) finSupp (0g𝐴))
5049anasss 467 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑈𝐺) finSupp (0g𝐴))
5148, 50eqbrtrrd 5103 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑙 ∈ ℕ0 ↦ (𝑈‘(𝐺𝑙))) finSupp (0g𝐴))
52 simpll1 1211 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → 𝑁 ∈ Fin)
5343ad2ant2 1133 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
5453ad2antrr 723 . . . . . . 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 2740 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘𝑃)
5856, 5, 10, 13, 57chpmatply1 21992 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐶𝑀) ∈ (Base‘𝑃))
5955, 58eqeltrid 2845 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐾 ∈ (Base‘𝑃))
60 eqid 2740 . . . . . . . . . 10 (coe1𝐾) = (coe1𝐾)
61 eqid 2740 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
6260, 57, 13, 61coe1fvalcl 21394 . . . . . . . . 9 ((𝐾 ∈ (Base‘𝑃) ∧ 𝑙 ∈ ℕ0) → ((coe1𝐾)‘𝑙) ∈ (Base‘𝑅))
6359, 62sylan 580 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑙 ∈ ℕ0) → ((coe1𝐾)‘𝑙) ∈ (Base‘𝑅))
6463adantlr 712 . . . . . . 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 19818 . . . . . . . . 9 (𝐴 ∈ Ring → 1𝐵)
678, 66syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 1𝐵)
6867ad2antrr 723 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → 1𝐵)
69 chcoeffeq.m . . . . . . . 8 = ( ·𝑠𝐴)
7061, 5, 10, 69matvscl 21591 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (((coe1𝐾)‘𝑙) ∈ (Base‘𝑅) ∧ 1𝐵)) → (((coe1𝐾)‘𝑙) 1 ) ∈ 𝐵)
7152, 54, 64, 68, 70syl22anc 836 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → (((coe1𝐾)‘𝑙) 1 ) ∈ 𝐵)
7271ralrimiva 3110 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ∀𝑙 ∈ ℕ0 (((coe1𝐾)‘𝑙) 1 ) ∈ 𝐵)
73 nn0ex 12250 . . . . . . 7 0 ∈ V
7473a1i 11 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ℕ0 ∈ V)
755matlmod 21589 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)
764, 75sylan2 593 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ LMod)
77763adant3 1131 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ LMod)
7877adantr 481 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝐴 ∈ LMod)
79 eqidd 2741 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (Scalar‘𝐴) = (Scalar‘𝐴))
80 fvexd 6786 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → ((coe1𝐾)‘𝑙) ∈ V)
81 eqid 2740 . . . . . 6 (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴))
825matsca2 21580 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝐴))
83823adant3 1131 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 = (Scalar‘𝐴))
8483, 53eqeltrrd 2842 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Scalar‘𝐴) ∈ Ring)
8583eqcomd 2746 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Scalar‘𝐴) = 𝑅)
8685fveq2d 6775 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Poly1‘(Scalar‘𝐴)) = (Poly1𝑅))
8786, 13eqtr4di 2798 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Poly1‘(Scalar‘𝐴)) = 𝑃)
8887fveq2d 6775 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Base‘(Poly1‘(Scalar‘𝐴))) = (Base‘𝑃))
8959, 88eleqtrrd 2844 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐾 ∈ (Base‘(Poly1‘(Scalar‘𝐴))))
90 eqid 2740 . . . . . . . . 9 (Poly1‘(Scalar‘𝐴)) = (Poly1‘(Scalar‘𝐴))
91 eqid 2740 . . . . . . . . 9 (Base‘(Poly1‘(Scalar‘𝐴))) = (Base‘(Poly1‘(Scalar‘𝐴)))
9290, 91, 81mptcoe1fsupp 21397 . . . . . . . 8 (((Scalar‘𝐴) ∈ Ring ∧ 𝐾 ∈ (Base‘(Poly1‘(Scalar‘𝐴)))) → (𝑙 ∈ ℕ0 ↦ ((coe1𝐾)‘𝑙)) finSupp (0g‘(Scalar‘𝐴)))
9384, 89, 92syl2anc 584 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑙 ∈ ℕ0 ↦ ((coe1𝐾)‘𝑙)) finSupp (0g‘(Scalar‘𝐴)))
9493adantr 481 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑙 ∈ ℕ0 ↦ ((coe1𝐾)‘𝑙)) finSupp (0g‘(Scalar‘𝐴)))
9574, 78, 79, 10, 80, 68, 12, 81, 69, 94mptscmfsupp0 20199 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑙 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑙) 1 )) finSupp (0g𝐴))
96 2fveq3 6776 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑈‘(𝐺𝑛)) = (𝑈‘(𝐺𝑙)))
97 oveq1 7279 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)) = (𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))
9896, 97oveq12d 7290 . . . . . . . 8 (𝑛 = 𝑙 → ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))) = ((𝑈‘(𝐺𝑙))( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
9998cbvmptv 5192 . . . . . . 7 (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))) = (𝑙 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑙))( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
10099oveq2i 7283 . . . . . 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)))))
101100a1i 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))))))
102 fveq2 6771 . . . . . . . . . 10 (𝑛 = 𝑙 → ((coe1𝐾)‘𝑛) = ((coe1𝐾)‘𝑙))
103102oveq1d 7287 . . . . . . . . 9 (𝑛 = 𝑙 → (((coe1𝐾)‘𝑛) 1 ) = (((coe1𝐾)‘𝑙) 1 ))
104103, 97oveq12d 7290 . . . . . . . 8 (𝑛 = 𝑙 → ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))) = ((((coe1𝐾)‘𝑙) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
105104cbvmptv 5192 . . . . . . 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))))
106105oveq2i 7283 . . . . . 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)))))
107106a1i 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))))))
1081, 2, 3, 9, 10, 11, 12, 41, 51, 72, 95, 101, 107gsumply1eq 21487 . . . 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 )))
109108biimpa 477 . . 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 ))
11096, 103eqeq12d 2756 . . . 4 (𝑛 = 𝑙 → ((𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ) ↔ (𝑈‘(𝐺𝑙)) = (((coe1𝐾)‘𝑙) 1 )))
111110cbvralvw 3381 . . 3 (∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ) ↔ ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺𝑙)) = (((coe1𝐾)‘𝑙) 1 ))
112109, 111sylibr 233 . 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 ))
113112ex 413 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 396  w3a 1086   = wceq 1542  wcel 2110  wral 3066  Vcvv 3431  ifcif 4465   class class class wbr 5079  cmpt 5162  ccom 5594  wf 6428  cfv 6432  (class class class)co 7272  m cmap 8607  Fincfn 8725   finSupp cfsupp 9116  0cc0 10882  1c1 10883   + caddc 10885   < clt 11020  cmin 11216  cn 11984  0cn0 12244  ...cfz 13250  Basecbs 16923  .rcmulr 16974  Scalarcsca 16976   ·𝑠 cvsca 16977  0gc0g 17161   Σg cgsu 17162  -gcsg 18590  .gcmg 18711  mulGrpcmgp 19731  1rcur 19748  Ringcrg 19794  CRingccrg 19795  LModclmod 20134  var1cv1 21358  Poly1cpl1 21359  coe1cco1 21360   Mat cmat 21565   maAdju cmadu 21792   ConstPolyMat ccpmat 21863   matToPolyMat cmat2pmat 21864   cPolyMatToMat ccpmat2mat 21865   CharPlyMat cchpmat 21986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7583  ax-cnex 10938  ax-resscn 10939  ax-1cn 10940  ax-icn 10941  ax-addcl 10942  ax-addrcl 10943  ax-mulcl 10944  ax-mulrcl 10945  ax-mulcom 10946  ax-addass 10947  ax-mulass 10948  ax-distr 10949  ax-i2m1 10950  ax-1ne0 10951  ax-1rid 10952  ax-rnegex 10953  ax-rrecex 10954  ax-cnre 10955  ax-pre-lttri 10956  ax-pre-lttrn 10957  ax-pre-ltadd 10958  ax-pre-mulgt0 10959  ax-addf 10961  ax-mulf 10962
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-xor 1507  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4846  df-int 4886  df-iun 4932  df-iin 4933  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-isom 6441  df-riota 7229  df-ov 7275  df-oprab 7276  df-mpo 7277  df-of 7528  df-ofr 7529  df-om 7708  df-1st 7825  df-2nd 7826  df-supp 7970  df-tpos 8034  df-frecs 8089  df-wrecs 8120  df-recs 8194  df-rdg 8233  df-1o 8289  df-2o 8290  df-er 8490  df-map 8609  df-pm 8610  df-ixp 8678  df-en 8726  df-dom 8727  df-sdom 8728  df-fin 8729  df-fsupp 9117  df-sup 9189  df-oi 9257  df-card 9708  df-pnf 11022  df-mnf 11023  df-xr 11024  df-ltxr 11025  df-le 11026  df-sub 11218  df-neg 11219  df-div 11644  df-nn 11985  df-2 12047  df-3 12048  df-4 12049  df-5 12050  df-6 12051  df-7 12052  df-8 12053  df-9 12054  df-n0 12245  df-xnn0 12317  df-z 12331  df-dec 12449  df-uz 12594  df-rp 12742  df-fz 13251  df-fzo 13394  df-seq 13733  df-exp 13794  df-hash 14056  df-word 14229  df-lsw 14277  df-concat 14285  df-s1 14312  df-substr 14365  df-pfx 14395  df-splice 14474  df-reverse 14483  df-s2 14572  df-struct 16859  df-sets 16876  df-slot 16894  df-ndx 16906  df-base 16924  df-ress 16953  df-plusg 16986  df-mulr 16987  df-starv 16988  df-sca 16989  df-vsca 16990  df-ip 16991  df-tset 16992  df-ple 16993  df-ds 16995  df-unif 16996  df-hom 16997  df-cco 16998  df-0g 17163  df-gsum 17164  df-prds 17169  df-pws 17171  df-mre 17306  df-mrc 17307  df-acs 17309  df-mgm 18337  df-sgrp 18386  df-mnd 18397  df-mhm 18441  df-submnd 18442  df-efmnd 18519  df-grp 18591  df-minusg 18592  df-sbg 18593  df-mulg 18712  df-subg 18763  df-ghm 18843  df-gim 18886  df-cntz 18934  df-oppg 18961  df-symg 18986  df-pmtr 19061  df-psgn 19110  df-cmn 19399  df-abl 19400  df-mgp 19732  df-ur 19749  df-srg 19753  df-ring 19796  df-cring 19797  df-oppr 19873  df-dvdsr 19894  df-unit 19895  df-invr 19925  df-dvr 19936  df-rnghom 19970  df-drng 20004  df-subrg 20033  df-lmod 20136  df-lss 20205  df-sra 20445  df-rgmod 20446  df-cnfld 20609  df-zring 20682  df-zrh 20716  df-dsmm 20950  df-frlm 20965  df-ascl 21073  df-psr 21123  df-mvr 21124  df-mpl 21125  df-opsr 21127  df-psr1 21362  df-vr1 21363  df-ply1 21364  df-coe1 21365  df-mamu 21544  df-mat 21566  df-mdet 21745  df-cpmat 21866  df-mat2pmat 21867  df-cpmat2mat 21868  df-chpmat 21987
This theorem is referenced by:  chcoeffeq  22046
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