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Theorem chcoeffeqlem 20969
Description: Lemma for chcoeffeq 20970. (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 2765 . . . . 5 (Poly1𝐴) = (Poly1𝐴)
2 eqid 2765 . . . . 5 (var1𝐴) = (var1𝐴)
3 eqid 2765 . . . . 5 (.g‘(mulGrp‘(Poly1𝐴))) = (.g‘(mulGrp‘(Poly1𝐴)))
4 crngring 18825 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5 chcoeffeq.a . . . . . . . . 9 𝐴 = (𝑁 Mat 𝑅)
65matring 20525 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
74, 6sylan2 586 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
873adant3 1162 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ Ring)
98adantr 472 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐴 ∈ Ring)
10 chcoeffeq.b . . . . 5 𝐵 = (Base‘𝐴)
11 eqid 2765 . . . . 5 ( ·𝑠 ‘(Poly1𝐴)) = ( ·𝑠 ‘(Poly1𝐴))
12 eqid 2765 . . . . 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 2765 . . . . . . . 8 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
21 eqid 2765 . . . . . . . 8 ( ·𝑠𝑌) = ( ·𝑠𝑌)
22 eqid 2765 . . . . . . . 8 (1r𝑌) = (1r𝑌)
23 eqid 2765 . . . . . . . 8 (var1𝑅) = (var1𝑅)
24 eqid 2765 . . . . . . . 8 (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) = (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))
25 eqid 2765 . . . . . . . 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 20965 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝑈𝐺) ∈ (𝐵𝑚0))
2928anasss 458 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑈𝐺) ∈ (𝐵𝑚0))
305, 10, 13, 14, 16, 17, 18, 15, 19, 20chfacfisfcpmat 20939 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
314, 30syl3anl2 1534 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
3231adantr 472 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
33 fvco3 6464 . . . . . . . . . 10 ((𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ0) → ((𝑈𝐺)‘𝑙) = (𝑈‘(𝐺𝑙)))
3433eqcomd 2771 . . . . . . . . 9 ((𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺𝑙)) = ((𝑈𝐺)‘𝑙))
3532, 34sylan 575 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺𝑙)) = ((𝑈𝐺)‘𝑙))
36 elmapi 8082 . . . . . . . . . 10 ((𝑈𝐺) ∈ (𝐵𝑚0) → (𝑈𝐺):ℕ0𝐵)
3736adantl 473 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) → (𝑈𝐺):ℕ0𝐵)
3837ffvelrnda 6549 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) ∧ 𝑙 ∈ ℕ0) → ((𝑈𝐺)‘𝑙) ∈ 𝐵)
3935, 38eqeltrd 2844 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺𝑙)) ∈ 𝐵)
4039ralrimiva 3113 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) → ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺𝑙)) ∈ 𝐵)
4129, 40mpdan 678 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺𝑙)) ∈ 𝐵)
424anim2i 610 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
43423adant3 1162 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
4443adantr 472 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
455, 10, 20, 27cpm2mf 20836 . . . . . . . 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 6591 . . . . . . 7 ((𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)) → (𝑈𝐺) = (𝑙 ∈ ℕ0 ↦ (𝑈‘(𝐺𝑙))))
4846, 31, 47syl2anc 579 . . . . . 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 20966 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝑈𝐺) finSupp (0g𝐴))
5049anasss 458 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑈𝐺) finSupp (0g𝐴))
5148, 50eqbrtrrd 4833 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑙 ∈ ℕ0 ↦ (𝑈‘(𝐺𝑙))) finSupp (0g𝐴))
52 simpll1 1269 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → 𝑁 ∈ Fin)
5343ad2ant2 1164 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
5453ad2antrr 717 . . . . . . 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 2765 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘𝑃)
5856, 5, 10, 13, 57chpmatply1 20916 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐶𝑀) ∈ (Base‘𝑃))
5955, 58syl5eqel 2848 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐾 ∈ (Base‘𝑃))
60 eqid 2765 . . . . . . . . . 10 (coe1𝐾) = (coe1𝐾)
61 eqid 2765 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
6260, 57, 13, 61coe1fvalcl 19855 . . . . . . . . 9 ((𝐾 ∈ (Base‘𝑃) ∧ 𝑙 ∈ ℕ0) → ((coe1𝐾)‘𝑙) ∈ (Base‘𝑅))
6359, 62sylan 575 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑙 ∈ ℕ0) → ((coe1𝐾)‘𝑙) ∈ (Base‘𝑅))
6463adantlr 706 . . . . . . 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 18835 . . . . . . . . 9 (𝐴 ∈ Ring → 1𝐵)
678, 66syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 1𝐵)
6867ad2antrr 717 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → 1𝐵)
69 chcoeffeq.m . . . . . . . 8 = ( ·𝑠𝐴)
7061, 5, 10, 69matvscl 20513 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (((coe1𝐾)‘𝑙) ∈ (Base‘𝑅) ∧ 1𝐵)) → (((coe1𝐾)‘𝑙) 1 ) ∈ 𝐵)
7152, 54, 64, 68, 70syl22anc 867 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → (((coe1𝐾)‘𝑙) 1 ) ∈ 𝐵)
7271ralrimiva 3113 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ∀𝑙 ∈ ℕ0 (((coe1𝐾)‘𝑙) 1 ) ∈ 𝐵)
73 nn0ex 11545 . . . . . . 7 0 ∈ V
7473a1i 11 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ℕ0 ∈ V)
755matlmod 20511 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)
764, 75sylan2 586 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ LMod)
77763adant3 1162 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ LMod)
7877adantr 472 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐴 ∈ LMod)
79 eqidd 2766 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (Scalar‘𝐴) = (Scalar‘𝐴))
80 fvexd 6390 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → ((coe1𝐾)‘𝑙) ∈ V)
81 eqid 2765 . . . . . 6 (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴))
825matsca2 20502 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝐴))
83823adant3 1162 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 = (Scalar‘𝐴))
8483, 53eqeltrrd 2845 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Scalar‘𝐴) ∈ Ring)
8583eqcomd 2771 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Scalar‘𝐴) = 𝑅)
8685fveq2d 6379 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Poly1‘(Scalar‘𝐴)) = (Poly1𝑅))
8786, 13syl6eqr 2817 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Poly1‘(Scalar‘𝐴)) = 𝑃)
8887fveq2d 6379 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Base‘(Poly1‘(Scalar‘𝐴))) = (Base‘𝑃))
8959, 88eleqtrrd 2847 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐾 ∈ (Base‘(Poly1‘(Scalar‘𝐴))))
90 eqid 2765 . . . . . . . . 9 (Poly1‘(Scalar‘𝐴)) = (Poly1‘(Scalar‘𝐴))
91 eqid 2765 . . . . . . . . 9 (Base‘(Poly1‘(Scalar‘𝐴))) = (Base‘(Poly1‘(Scalar‘𝐴)))
9290, 91, 81mptcoe1fsupp 19858 . . . . . . . 8 (((Scalar‘𝐴) ∈ Ring ∧ 𝐾 ∈ (Base‘(Poly1‘(Scalar‘𝐴)))) → (𝑙 ∈ ℕ0 ↦ ((coe1𝐾)‘𝑙)) finSupp (0g‘(Scalar‘𝐴)))
9384, 89, 92syl2anc 579 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑙 ∈ ℕ0 ↦ ((coe1𝐾)‘𝑙)) finSupp (0g‘(Scalar‘𝐴)))
9493adantr 472 . . . . . 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 19197 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑙 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑙) 1 )) finSupp (0g𝐴))
96 2fveq3 6380 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑈‘(𝐺𝑛)) = (𝑈‘(𝐺𝑙)))
97 oveq1 6849 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)) = (𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))
9896, 97oveq12d 6860 . . . . . . . 8 (𝑛 = 𝑙 → ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))) = ((𝑈‘(𝐺𝑙))( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
9998cbvmptv 4909 . . . . . . 7 (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))) = (𝑙 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑙))( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
10099oveq2i 6853 . . . . . 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 6375 . . . . . . . . . 10 (𝑛 = 𝑙 → ((coe1𝐾)‘𝑛) = ((coe1𝐾)‘𝑙))
103102oveq1d 6857 . . . . . . . . 9 (𝑛 = 𝑙 → (((coe1𝐾)‘𝑛) 1 ) = (((coe1𝐾)‘𝑙) 1 ))
104103, 97oveq12d 6860 . . . . . . . 8 (𝑛 = 𝑙 → ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))) = ((((coe1𝐾)‘𝑙) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
105104cbvmptv 4909 . . . . . . 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 6853 . . . . . 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 19948 . . . 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 468 . . 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 2780 . . . 4 (𝑛 = 𝑙 → ((𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ) ↔ (𝑈‘(𝐺𝑙)) = (((coe1𝐾)‘𝑙) 1 )))
111110cbvralv 3319 . . 3 (∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ) ↔ ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺𝑙)) = (((coe1𝐾)‘𝑙) 1 ))
112109, 111sylibr 225 . 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 401 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 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  ifcif 4243   class class class wbr 4809  cmpt 4888  ccom 5281  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  Fincfn 8160   finSupp cfsupp 8482  0cc0 10189  1c1 10190   + caddc 10192   < clt 10328  cmin 10520  cn 11274  0cn0 11538  ...cfz 12533  Basecbs 16132  .rcmulr 16217  Scalarcsca 16219   ·𝑠 cvsca 16220  0gc0g 16368   Σg cgsu 16369  -gcsg 17693  .gcmg 17809  mulGrpcmgp 18756  1rcur 18768  Ringcrg 18814  CRingccrg 18815  LModclmod 19132  var1cv1 19819  Poly1cpl1 19820  coe1cco1 19821   Mat cmat 20489   maAdju cmadu 20715   ConstPolyMat ccpmat 20787   matToPolyMat cmat2pmat 20788   cPolyMatToMat ccpmat2mat 20789   CharPlyMat cchpmat 20910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-addf 10268  ax-mulf 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-xor 1634  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-ot 4343  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-ofr 7096  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-tpos 7555  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-sup 8555  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-xnn0 11611  df-z 11625  df-dec 11741  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-word 13487  df-lsw 13534  df-concat 13542  df-s1 13567  df-substr 13617  df-pfx 13662  df-splice 13765  df-reverse 13783  df-s2 13879  df-struct 16134  df-ndx 16135  df-slot 16136  df-base 16138  df-sets 16139  df-ress 16140  df-plusg 16229  df-mulr 16230  df-starv 16231  df-sca 16232  df-vsca 16233  df-ip 16234  df-tset 16235  df-ple 16236  df-ds 16238  df-unif 16239  df-hom 16240  df-cco 16241  df-0g 16370  df-gsum 16371  df-prds 16376  df-pws 16378  df-mre 16514  df-mrc 16515  df-acs 16517  df-mgm 17510  df-sgrp 17552  df-mnd 17563  df-mhm 17603  df-submnd 17604  df-grp 17694  df-minusg 17695  df-sbg 17696  df-mulg 17810  df-subg 17857  df-ghm 17924  df-gim 17967  df-cntz 18015  df-oppg 18041  df-symg 18063  df-pmtr 18127  df-psgn 18176  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-srg 18773  df-ring 18816  df-cring 18817  df-oppr 18890  df-dvdsr 18908  df-unit 18909  df-invr 18939  df-dvr 18950  df-rnghom 18984  df-drng 19018  df-subrg 19047  df-lmod 19134  df-lss 19202  df-sra 19446  df-rgmod 19447  df-ascl 19588  df-psr 19630  df-mvr 19631  df-mpl 19632  df-opsr 19634  df-psr1 19823  df-vr1 19824  df-ply1 19825  df-coe1 19826  df-cnfld 20020  df-zring 20092  df-zrh 20125  df-dsmm 20352  df-frlm 20367  df-mamu 20466  df-mat 20490  df-mdet 20668  df-cpmat 20790  df-mat2pmat 20791  df-cpmat2mat 20792  df-chpmat 20911
This theorem is referenced by:  chcoeffeq  20970
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