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Theorem chcoeffeqlem 22034
Description: Lemma for chcoeffeq 22035. (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 2738 . . . . 5 (Poly1𝐴) = (Poly1𝐴)
2 eqid 2738 . . . . 5 (var1𝐴) = (var1𝐴)
3 eqid 2738 . . . . 5 (.g‘(mulGrp‘(Poly1𝐴))) = (.g‘(mulGrp‘(Poly1𝐴)))
4 crngring 19795 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5 chcoeffeq.a . . . . . . . . 9 𝐴 = (𝑁 Mat 𝑅)
65matring 21592 . . . . . . . 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 2738 . . . . 5 ( ·𝑠 ‘(Poly1𝐴)) = ( ·𝑠 ‘(Poly1𝐴))
12 eqid 2738 . . . . 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 2738 . . . . . . . 8 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
21 eqid 2738 . . . . . . . 8 ( ·𝑠𝑌) = ( ·𝑠𝑌)
22 eqid 2738 . . . . . . . 8 (1r𝑌) = (1r𝑌)
23 eqid 2738 . . . . . . . 8 (var1𝑅) = (var1𝑅)
24 eqid 2738 . . . . . . . 8 (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) = (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))
25 eqid 2738 . . . . . . . 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 22030 . . . . . . 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 22004 . . . . . . . . . . 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 6867 . . . . . . . . . 10 ((𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ0) → ((𝑈𝐺)‘𝑙) = (𝑈‘(𝐺𝑙)))
3433eqcomd 2744 . . . . . . . . 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 8637 . . . . . . . . . 10 ((𝑈𝐺) ∈ (𝐵m0) → (𝑈𝐺):ℕ0𝐵)
3736adantl 482 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵m0)) → (𝑈𝐺):ℕ0𝐵)
3837ffvelrnda 6961 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵m0)) ∧ 𝑙 ∈ ℕ0) → ((𝑈𝐺)‘𝑙) ∈ 𝐵)
3935, 38eqeltrd 2839 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵m0)) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺𝑙)) ∈ 𝐵)
4039ralrimiva 3103 . . . . . 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 21901 . . . . . . . 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 7005 . . . . . . 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 22031 . . . . . . 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 5098 . . . . 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 2738 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘𝑃)
5856, 5, 10, 13, 57chpmatply1 21981 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐶𝑀) ∈ (Base‘𝑃))
5955, 58eqeltrid 2843 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐾 ∈ (Base‘𝑃))
60 eqid 2738 . . . . . . . . . 10 (coe1𝐾) = (coe1𝐾)
61 eqid 2738 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
6260, 57, 13, 61coe1fvalcl 21383 . . . . . . . . 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 19807 . . . . . . . . 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 21580 . . . . . . 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 3103 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ∀𝑙 ∈ ℕ0 (((coe1𝐾)‘𝑙) 1 ) ∈ 𝐵)
73 nn0ex 12239 . . . . . . 7 0 ∈ V
7473a1i 11 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ℕ0 ∈ V)
755matlmod 21578 . . . . . . . . 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 2739 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (Scalar‘𝐴) = (Scalar‘𝐴))
80 fvexd 6789 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → ((coe1𝐾)‘𝑙) ∈ V)
81 eqid 2738 . . . . . 6 (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴))
825matsca2 21569 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝐴))
83823adant3 1131 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 = (Scalar‘𝐴))
8483, 53eqeltrrd 2840 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Scalar‘𝐴) ∈ Ring)
8583eqcomd 2744 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Scalar‘𝐴) = 𝑅)
8685fveq2d 6778 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Poly1‘(Scalar‘𝐴)) = (Poly1𝑅))
8786, 13eqtr4di 2796 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Poly1‘(Scalar‘𝐴)) = 𝑃)
8887fveq2d 6778 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Base‘(Poly1‘(Scalar‘𝐴))) = (Base‘𝑃))
8959, 88eleqtrrd 2842 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐾 ∈ (Base‘(Poly1‘(Scalar‘𝐴))))
90 eqid 2738 . . . . . . . . 9 (Poly1‘(Scalar‘𝐴)) = (Poly1‘(Scalar‘𝐴))
91 eqid 2738 . . . . . . . . 9 (Base‘(Poly1‘(Scalar‘𝐴))) = (Base‘(Poly1‘(Scalar‘𝐴)))
9290, 91, 81mptcoe1fsupp 21386 . . . . . . . 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 20188 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑙 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑙) 1 )) finSupp (0g𝐴))
96 2fveq3 6779 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑈‘(𝐺𝑛)) = (𝑈‘(𝐺𝑙)))
97 oveq1 7282 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)) = (𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))
9896, 97oveq12d 7293 . . . . . . . 8 (𝑛 = 𝑙 → ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))) = ((𝑈‘(𝐺𝑙))( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
9998cbvmptv 5187 . . . . . . 7 (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))) = (𝑙 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑙))( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
10099oveq2i 7286 . . . . . 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 6774 . . . . . . . . . 10 (𝑛 = 𝑙 → ((coe1𝐾)‘𝑛) = ((coe1𝐾)‘𝑙))
103102oveq1d 7290 . . . . . . . . 9 (𝑛 = 𝑙 → (((coe1𝐾)‘𝑛) 1 ) = (((coe1𝐾)‘𝑙) 1 ))
104103, 97oveq12d 7293 . . . . . . . 8 (𝑛 = 𝑙 → ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))) = ((((coe1𝐾)‘𝑙) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
105104cbvmptv 5187 . . . . . . 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 7286 . . . . . 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 21476 . . . 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 2754 . . . 4 (𝑛 = 𝑙 → ((𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ) ↔ (𝑈‘(𝐺𝑙)) = (((coe1𝐾)‘𝑙) 1 )))
111110cbvralvw 3383 . . 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 1539  wcel 2106  wral 3064  Vcvv 3432  ifcif 4459   class class class wbr 5074  cmpt 5157  ccom 5593  wf 6429  cfv 6433  (class class class)co 7275  m cmap 8615  Fincfn 8733   finSupp cfsupp 9128  0cc0 10871  1c1 10872   + caddc 10874   < clt 11009  cmin 11205  cn 11973  0cn0 12233  ...cfz 13239  Basecbs 16912  .rcmulr 16963  Scalarcsca 16965   ·𝑠 cvsca 16966  0gc0g 17150   Σg cgsu 17151  -gcsg 18579  .gcmg 18700  mulGrpcmgp 19720  1rcur 19737  Ringcrg 19783  CRingccrg 19784  LModclmod 20123  var1cv1 21347  Poly1cpl1 21348  coe1cco1 21349   Mat cmat 21554   maAdju cmadu 21781   ConstPolyMat ccpmat 21852   matToPolyMat cmat2pmat 21853   cPolyMatToMat ccpmat2mat 21854   CharPlyMat cchpmat 21975
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-addf 10950  ax-mulf 10951
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 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-ot 4570  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-ofr 7534  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-tpos 8042  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-sup 9201  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-xnn0 12306  df-z 12320  df-dec 12438  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-word 14218  df-lsw 14266  df-concat 14274  df-s1 14301  df-substr 14354  df-pfx 14384  df-splice 14463  df-reverse 14472  df-s2 14561  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-0g 17152  df-gsum 17153  df-prds 17158  df-pws 17160  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-efmnd 18508  df-grp 18580  df-minusg 18581  df-sbg 18582  df-mulg 18701  df-subg 18752  df-ghm 18832  df-gim 18875  df-cntz 18923  df-oppg 18950  df-symg 18975  df-pmtr 19050  df-psgn 19099  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-srg 19742  df-ring 19785  df-cring 19786  df-oppr 19862  df-dvdsr 19883  df-unit 19884  df-invr 19914  df-dvr 19925  df-rnghom 19959  df-drng 19993  df-subrg 20022  df-lmod 20125  df-lss 20194  df-sra 20434  df-rgmod 20435  df-cnfld 20598  df-zring 20671  df-zrh 20705  df-dsmm 20939  df-frlm 20954  df-ascl 21062  df-psr 21112  df-mvr 21113  df-mpl 21114  df-opsr 21116  df-psr1 21351  df-vr1 21352  df-ply1 21353  df-coe1 21354  df-mamu 21533  df-mat 21555  df-mdet 21734  df-cpmat 21855  df-mat2pmat 21856  df-cpmat2mat 21857  df-chpmat 21976
This theorem is referenced by:  chcoeffeq  22035
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