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Theorem chcoeffeqlem 20813
Description: Lemma for chcoeffeq 20814. (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 2724 . . . . 5 (Poly1𝐴) = (Poly1𝐴)
2 eqid 2724 . . . . 5 (var1𝐴) = (var1𝐴)
3 eqid 2724 . . . . 5 (.g‘(mulGrp‘(Poly1𝐴))) = (.g‘(mulGrp‘(Poly1𝐴)))
4 crngring 18679 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5 chcoeffeq.a . . . . . . . . 9 𝐴 = (𝑁 Mat 𝑅)
65matring 20372 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
74, 6sylan2 492 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
873adant3 1124 . . . . . 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 2724 . . . . 5 ( ·𝑠 ‘(Poly1𝐴)) = ( ·𝑠 ‘(Poly1𝐴))
12 eqid 2724 . . . . 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 2724 . . . . . . . 8 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
21 eqid 2724 . . . . . . . 8 ( ·𝑠𝑌) = ( ·𝑠𝑌)
22 eqid 2724 . . . . . . . 8 (1r𝑌) = (1r𝑌)
23 eqid 2724 . . . . . . . 8 (var1𝑅) = (var1𝑅)
24 eqid 2724 . . . . . . . 8 (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) = (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))
25 eqid 2724 . . . . . . . 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 20809 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝑈𝐺) ∈ (𝐵𝑚0))
2928anasss 682 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑈𝐺) ∈ (𝐵𝑚0))
305, 10, 13, 14, 16, 17, 18, 15, 19, 20chfacfisfcpmat 20783 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
314, 30syl3anl2 1488 . . . . . . . . . 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 6389 . . . . . . . . . 10 ((𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ0) → ((𝑈𝐺)‘𝑙) = (𝑈‘(𝐺𝑙)))
3433eqcomd 2730 . . . . . . . . 9 ((𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺𝑙)) = ((𝑈𝐺)‘𝑙))
3532, 34sylan 489 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺𝑙)) = ((𝑈𝐺)‘𝑙))
36 elmapi 7996 . . . . . . . . . 10 ((𝑈𝐺) ∈ (𝐵𝑚0) → (𝑈𝐺):ℕ0𝐵)
3736adantl 473 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) → (𝑈𝐺):ℕ0𝐵)
3837ffvelrnda 6474 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) ∧ 𝑙 ∈ ℕ0) → ((𝑈𝐺)‘𝑙) ∈ 𝐵)
3935, 38eqeltrd 2803 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺𝑙)) ∈ 𝐵)
4039ralrimiva 3068 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑈𝐺) ∈ (𝐵𝑚0)) → ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺𝑙)) ∈ 𝐵)
4129, 40mpdan 705 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺𝑙)) ∈ 𝐵)
424anim2i 594 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
43423adant3 1124 . . . . . . . . 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 20680 . . . . . . . 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 6515 . . . . . . 7 ((𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)) → (𝑈𝐺) = (𝑙 ∈ ℕ0 ↦ (𝑈‘(𝐺𝑙))))
4846, 31, 47syl2anc 696 . . . . . 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 20810 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝑈𝐺) finSupp (0g𝐴))
5049anasss 682 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑈𝐺) finSupp (0g𝐴))
5148, 50eqbrtrrd 4784 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑙 ∈ ℕ0 ↦ (𝑈‘(𝐺𝑙))) finSupp (0g𝐴))
52 simpll1 1231 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → 𝑁 ∈ Fin)
5343ad2ant2 1126 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
5453ad2antrr 764 . . . . . . 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 2724 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘𝑃)
5856, 5, 10, 13, 57chpmatply1 20760 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐶𝑀) ∈ (Base‘𝑃))
5955, 58syl5eqel 2807 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐾 ∈ (Base‘𝑃))
60 eqid 2724 . . . . . . . . . 10 (coe1𝐾) = (coe1𝐾)
61 eqid 2724 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
6260, 57, 13, 61coe1fvalcl 19705 . . . . . . . . 9 ((𝐾 ∈ (Base‘𝑃) ∧ 𝑙 ∈ ℕ0) → ((coe1𝐾)‘𝑙) ∈ (Base‘𝑅))
6359, 62sylan 489 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑙 ∈ ℕ0) → ((coe1𝐾)‘𝑙) ∈ (Base‘𝑅))
6463adantlr 753 . . . . . . 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 18689 . . . . . . . . 9 (𝐴 ∈ Ring → 1𝐵)
678, 66syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 1𝐵)
6867ad2antrr 764 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → 1𝐵)
69 chcoeffeq.m . . . . . . . 8 = ( ·𝑠𝐴)
7061, 5, 10, 69matvscl 20360 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (((coe1𝐾)‘𝑙) ∈ (Base‘𝑅) ∧ 1𝐵)) → (((coe1𝐾)‘𝑙) 1 ) ∈ 𝐵)
7152, 54, 64, 68, 70syl22anc 1440 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → (((coe1𝐾)‘𝑙) 1 ) ∈ 𝐵)
7271ralrimiva 3068 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ∀𝑙 ∈ ℕ0 (((coe1𝐾)‘𝑙) 1 ) ∈ 𝐵)
73 nn0ex 11411 . . . . . . 7 0 ∈ V
7473a1i 11 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ℕ0 ∈ V)
755matlmod 20358 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)
764, 75sylan2 492 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ LMod)
77763adant3 1124 . . . . . . 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 2725 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (Scalar‘𝐴) = (Scalar‘𝐴))
80 fvexd 6316 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → ((coe1𝐾)‘𝑙) ∈ V)
81 eqid 2724 . . . . . 6 (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴))
825matsca2 20349 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝐴))
83823adant3 1124 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 = (Scalar‘𝐴))
8483, 53eqeltrrd 2804 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Scalar‘𝐴) ∈ Ring)
8583eqcomd 2730 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Scalar‘𝐴) = 𝑅)
8685fveq2d 6308 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Poly1‘(Scalar‘𝐴)) = (Poly1𝑅))
8786, 13syl6eqr 2776 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Poly1‘(Scalar‘𝐴)) = 𝑃)
8887fveq2d 6308 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Base‘(Poly1‘(Scalar‘𝐴))) = (Base‘𝑃))
8959, 88eleqtrrd 2806 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐾 ∈ (Base‘(Poly1‘(Scalar‘𝐴))))
90 eqid 2724 . . . . . . . . 9 (Poly1‘(Scalar‘𝐴)) = (Poly1‘(Scalar‘𝐴))
91 eqid 2724 . . . . . . . . 9 (Base‘(Poly1‘(Scalar‘𝐴))) = (Base‘(Poly1‘(Scalar‘𝐴)))
9290, 91, 81mptcoe1fsupp 19708 . . . . . . . 8 (((Scalar‘𝐴) ∈ Ring ∧ 𝐾 ∈ (Base‘(Poly1‘(Scalar‘𝐴)))) → (𝑙 ∈ ℕ0 ↦ ((coe1𝐾)‘𝑙)) finSupp (0g‘(Scalar‘𝐴)))
9384, 89, 92syl2anc 696 . . . . . . 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 19051 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑙 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑙) 1 )) finSupp (0g𝐴))
96 fveq2 6304 . . . . . . . . . 10 (𝑛 = 𝑙 → (𝐺𝑛) = (𝐺𝑙))
9796fveq2d 6308 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑈‘(𝐺𝑛)) = (𝑈‘(𝐺𝑙)))
98 oveq1 6772 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)) = (𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))
9997, 98oveq12d 6783 . . . . . . . 8 (𝑛 = 𝑙 → ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))) = ((𝑈‘(𝐺𝑙))( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
10099cbvmptv 4858 . . . . . . 7 (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))) = (𝑙 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑙))( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
101100oveq2i 6776 . . . . . 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)))))
102101a1i 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))))))
103 fveq2 6304 . . . . . . . . . 10 (𝑛 = 𝑙 → ((coe1𝐾)‘𝑛) = ((coe1𝐾)‘𝑙))
104103oveq1d 6780 . . . . . . . . 9 (𝑛 = 𝑙 → (((coe1𝐾)‘𝑛) 1 ) = (((coe1𝐾)‘𝑙) 1 ))
105104, 98oveq12d 6783 . . . . . . . 8 (𝑛 = 𝑙 → ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))) = ((((coe1𝐾)‘𝑙) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑙(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))
106105cbvmptv 4858 . . . . . . 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))))
107106oveq2i 6776 . . . . . 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)))))
108107a1i 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))))))
1091, 2, 3, 9, 10, 11, 12, 41, 51, 72, 95, 102, 108gsumply1eq 19798 . . . 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 )))
110109biimpa 502 . . 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 ))
11197, 104eqeq12d 2739 . . . 4 (𝑛 = 𝑙 → ((𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ) ↔ (𝑈‘(𝐺𝑙)) = (((coe1𝐾)‘𝑙) 1 )))
112111cbvralv 3274 . . 3 (∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ) ↔ ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺𝑙)) = (((coe1𝐾)‘𝑙) 1 ))
113110, 112sylibr 224 . 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 ))
114113ex 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 1072   = wceq 1596  wcel 2103  wral 3014  Vcvv 3304  ifcif 4194   class class class wbr 4760  cmpt 4837  ccom 5222  wf 5997  cfv 6001  (class class class)co 6765  𝑚 cmap 7974  Fincfn 8072   finSupp cfsupp 8391  0cc0 10049  1c1 10050   + caddc 10052   < clt 10187  cmin 10379  cn 11133  0cn0 11405  ...cfz 12440  Basecbs 15980  .rcmulr 16065  Scalarcsca 16067   ·𝑠 cvsca 16068  0gc0g 16223   Σg cgsu 16224  -gcsg 17546  .gcmg 17662  mulGrpcmgp 18610  1rcur 18622  Ringcrg 18668  CRingccrg 18669  LModclmod 18986  var1cv1 19669  Poly1cpl1 19670  coe1cco1 19671   Mat cmat 20336   maAdju cmadu 20561   ConstPolyMat ccpmat 20631   matToPolyMat cmat2pmat 20632   cPolyMatToMat ccpmat2mat 20633   CharPlyMat cchpmat 20754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126  ax-addf 10128  ax-mulf 10129
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-xor 1578  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-ot 4294  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-of 7014  df-ofr 7015  df-om 7183  df-1st 7285  df-2nd 7286  df-supp 7416  df-tpos 7472  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-2o 7681  df-oadd 7684  df-er 7862  df-map 7976  df-pm 7977  df-ixp 8026  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-fsupp 8392  df-sup 8464  df-oi 8531  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-div 10798  df-nn 11134  df-2 11192  df-3 11193  df-4 11194  df-5 11195  df-6 11196  df-7 11197  df-8 11198  df-9 11199  df-n0 11406  df-xnn0 11477  df-z 11491  df-dec 11607  df-uz 11801  df-rp 11947  df-fz 12441  df-fzo 12581  df-seq 12917  df-exp 12976  df-hash 13233  df-word 13406  df-lsw 13407  df-concat 13408  df-s1 13409  df-substr 13410  df-splice 13411  df-reverse 13412  df-s2 13714  df-struct 15982  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-ress 15988  df-plusg 16077  df-mulr 16078  df-starv 16079  df-sca 16080  df-vsca 16081  df-ip 16082  df-tset 16083  df-ple 16084  df-ds 16087  df-unif 16088  df-hom 16089  df-cco 16090  df-0g 16225  df-gsum 16226  df-prds 16231  df-pws 16233  df-mre 16369  df-mrc 16370  df-acs 16372  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-mhm 17457  df-submnd 17458  df-grp 17547  df-minusg 17548  df-sbg 17549  df-mulg 17663  df-subg 17713  df-ghm 17780  df-gim 17823  df-cntz 17871  df-oppg 17897  df-symg 17919  df-pmtr 17983  df-psgn 18032  df-cmn 18316  df-abl 18317  df-mgp 18611  df-ur 18623  df-srg 18627  df-ring 18670  df-cring 18671  df-oppr 18744  df-dvdsr 18762  df-unit 18763  df-invr 18793  df-dvr 18804  df-rnghom 18838  df-drng 18872  df-subrg 18901  df-lmod 18988  df-lss 19056  df-sra 19295  df-rgmod 19296  df-ascl 19437  df-psr 19479  df-mvr 19480  df-mpl 19481  df-opsr 19483  df-psr1 19673  df-vr1 19674  df-ply1 19675  df-coe1 19676  df-cnfld 19870  df-zring 19942  df-zrh 19975  df-dsmm 20199  df-frlm 20214  df-mamu 20313  df-mat 20337  df-mdet 20514  df-cpmat 20634  df-mat2pmat 20635  df-cpmat2mat 20636  df-chpmat 20755
This theorem is referenced by:  chcoeffeq  20814
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