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Theorem cayhamlem4 22237
Description: Lemma for cayleyhamilton 22239. (Contributed by AV, 25-Nov-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)
cayhamlem.e1 = (.g‘(mulGrp‘𝐴))
cayhamlem.e2 𝐸 = (.g‘(mulGrp‘𝑌))
Assertion
Ref Expression
cayhamlem4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑛) (𝑛 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))))
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,𝑠,𝐴   𝐵,𝑏,𝑠   𝑀,𝑏,𝑠   𝑁,𝑏,𝑠   𝑃,𝑏,𝑛,𝑠   𝑅,𝑏,𝑠   𝑇,𝑏,𝑛,𝑠   𝑛,𝑊   𝑌,𝑏,𝑠   0 ,𝑛   × ,𝑛   ,𝑏,𝑛,𝑠   ,𝑛
Allowed substitution hints:   𝐶(𝑛,𝑠,𝑏)   × (𝑠,𝑏)   𝑈(𝑠,𝑏)   1 (𝑠,𝑏)   𝐸(𝑛,𝑠,𝑏)   (𝑠,𝑏)   𝐺(𝑠,𝑏)   (𝑠,𝑏)   𝐾(𝑠,𝑏)   𝑊(𝑠,𝑏)   0 (𝑠,𝑏)

Proof of Theorem cayhamlem4
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑛) (𝑛 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑛) (𝑛 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))))
2 simp1 1136 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑁 ∈ Fin)
32ad2antrr 724 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑁 ∈ Fin)
4 crngring 19976 . . . . . . 7 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
543ad2ant2 1134 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
65ad2antrr 724 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑅 ∈ Ring)
7 chcoeffeq.b . . . . . 6 𝐵 = (Base‘𝐴)
8 eqid 2736 . . . . . 6 (0g𝐴) = (0g𝐴)
9 chcoeffeq.a . . . . . . . . . . 11 𝐴 = (𝑁 Mat 𝑅)
109matring 21792 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
114, 10sylan2 593 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
12 ringcmn 20003 . . . . . . . . 9 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
1311, 12syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ CMnd)
14133adant3 1132 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ CMnd)
1514ad2antrr 724 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝐴 ∈ CMnd)
16 nn0ex 12419 . . . . . . 7 0 ∈ V
1716a1i 11 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → ℕ0 ∈ V)
183, 6, 10syl2anc 584 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝐴 ∈ Ring)
1918adantr 481 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
20 eqid 2736 . . . . . . . . . 10 (mulGrp‘𝐴) = (mulGrp‘𝐴)
2120, 7mgpbas 19902 . . . . . . . . 9 𝐵 = (Base‘(mulGrp‘𝐴))
22 cayhamlem.e1 . . . . . . . . 9 = (.g‘(mulGrp‘𝐴))
232, 5, 10syl2anc 584 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ Ring)
2420ringmgp 19970 . . . . . . . . . . 11 (𝐴 ∈ Ring → (mulGrp‘𝐴) ∈ Mnd)
2523, 24syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (mulGrp‘𝐴) ∈ Mnd)
2625ad3antrrr 728 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (mulGrp‘𝐴) ∈ Mnd)
27 simpr 485 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
28 simpll3 1214 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑀𝐵)
2928adantr 481 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑀𝐵)
3021, 22, 26, 27, 29mulgnn0cld 18897 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑀) ∈ 𝐵)
31 eqid 2736 . . . . . . . . . . . 12 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
32 chcoeffeq.u . . . . . . . . . . . 12 𝑈 = (𝑁 cPolyMatToMat 𝑅)
339, 7, 31, 32cpm2mf 22101 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
342, 5, 33syl2anc 584 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
3534ad3antrrr 728 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
36 simplr 767 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑠 ∈ ℕ)
37 simpr 485 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑏 ∈ (𝐵m (0...𝑠)))
38 chcoeffeq.p . . . . . . . . . . . 12 𝑃 = (Poly1𝑅)
39 chcoeffeq.y . . . . . . . . . . . 12 𝑌 = (𝑁 Mat 𝑃)
40 chcoeffeq.r . . . . . . . . . . . 12 × = (.r𝑌)
41 chcoeffeq.s . . . . . . . . . . . 12 = (-g𝑌)
42 chcoeffeq.0 . . . . . . . . . . . 12 0 = (0g𝑌)
43 chcoeffeq.t . . . . . . . . . . . 12 𝑇 = (𝑁 matToPolyMat 𝑅)
44 chcoeffeq.g . . . . . . . . . . . 12 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))))))))
459, 7, 38, 39, 40, 41, 42, 43, 44, 31chfacfisfcpmat 22204 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
463, 6, 28, 36, 37, 45syl32anc 1378 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
4746ffvelcdmda 7035 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (𝑁 ConstPolyMat 𝑅))
4835, 47ffvelcdmd 7036 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺𝑛)) ∈ 𝐵)
49 eqid 2736 . . . . . . . . 9 (.r𝐴) = (.r𝐴)
507, 49ringcl 19981 . . . . . . . 8 ((𝐴 ∈ Ring ∧ (𝑛 𝑀) ∈ 𝐵 ∧ (𝑈‘(𝐺𝑛)) ∈ 𝐵) → ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ 𝐵)
5119, 30, 48, 50syl3anc 1371 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ 𝐵)
5251fmpttd 7063 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))):ℕ0𝐵)
53 fvexd 6857 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (0g𝐴) ∈ V)
54 ovexd 7392 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ V)
559, 7, 38, 39, 40, 41, 42, 43, 44chfacffsupp 22205 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝐺 finSupp (0g𝑌))
5655anassrs 468 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝐺 finSupp (0g𝑌))
57 ovex 7390 . . . . . . . . . . . . 13 (𝑁 ConstPolyMat 𝑅) ∈ V
5857, 16pm3.2i 471 . . . . . . . . . . . 12 ((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ0 ∈ V)
59 elmapg 8778 . . . . . . . . . . . 12 (((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ0 ∈ V) → (𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m0) ↔ 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)))
6058, 59mp1i 13 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m0) ↔ 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)))
6146, 60mpbird 256 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m0))
62 fvex 6855 . . . . . . . . . 10 (0g𝑌) ∈ V
63 fsuppmapnn0ub 13900 . . . . . . . . . 10 ((𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m0) ∧ (0g𝑌) ∈ V) → (𝐺 finSupp (0g𝑌) → ∃𝑤 ∈ ℕ0𝑧 ∈ ℕ0 (𝑤 < 𝑧 → (𝐺𝑧) = (0g𝑌))))
6461, 62, 63sylancl 586 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝐺 finSupp (0g𝑌) → ∃𝑤 ∈ ℕ0𝑧 ∈ ℕ0 (𝑤 < 𝑧 → (𝐺𝑧) = (0g𝑌))))
65 csbov12g 7401 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (𝑧 / 𝑛(𝑛 𝑀)(.r𝐴)𝑧 / 𝑛(𝑈‘(𝐺𝑛))))
66 csbov1g 7402 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑛 𝑀) = (𝑧 / 𝑛𝑛 𝑀))
67 csbvarg 4391 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℕ0𝑧 / 𝑛𝑛 = 𝑧)
6867oveq1d 7372 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0 → (𝑧 / 𝑛𝑛 𝑀) = (𝑧 𝑀))
6966, 68eqtrd 2776 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑛 𝑀) = (𝑧 𝑀))
70 csbfv2g 6891 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑈‘(𝐺𝑛)) = (𝑈𝑧 / 𝑛(𝐺𝑛)))
71 csbfv 6892 . . . . . . . . . . . . . . . . . . . . 21 𝑧 / 𝑛(𝐺𝑛) = (𝐺𝑧)
7271a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝐺𝑛) = (𝐺𝑧))
7372fveq2d 6846 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0 → (𝑈𝑧 / 𝑛(𝐺𝑛)) = (𝑈‘(𝐺𝑧)))
7470, 73eqtrd 2776 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑈‘(𝐺𝑛)) = (𝑈‘(𝐺𝑧)))
7569, 74oveq12d 7375 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0 → (𝑧 / 𝑛(𝑛 𝑀)(.r𝐴)𝑧 / 𝑛(𝑈‘(𝐺𝑛))) = ((𝑧 𝑀)(.r𝐴)(𝑈‘(𝐺𝑧))))
7665, 75eqtrd 2776 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ℕ0𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = ((𝑧 𝑀)(.r𝐴)(𝑈‘(𝐺𝑧))))
7776ad2antlr 725 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → 𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = ((𝑧 𝑀)(.r𝐴)(𝑈‘(𝐺𝑧))))
78 fveq2 6842 . . . . . . . . . . . . . . . . 17 ((𝐺𝑧) = (0g𝑌) → (𝑈‘(𝐺𝑧)) = (𝑈‘(0g𝑌)))
792, 5jca 512 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
8079adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
81 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (0g𝑌) = (0g𝑌)
829, 32, 38, 39, 8, 81m2cpminv0 22110 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑈‘(0g𝑌)) = (0g𝐴))
8380, 82syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) → (𝑈‘(0g𝑌)) = (0g𝐴))
8483ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝑈‘(0g𝑌)) = (0g𝐴))
8578, 84sylan9eqr 2798 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → (𝑈‘(𝐺𝑧)) = (0g𝐴))
8685oveq2d 7373 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → ((𝑧 𝑀)(.r𝐴)(𝑈‘(𝐺𝑧))) = ((𝑧 𝑀)(.r𝐴)(0g𝐴)))
8718adantr 481 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝐴 ∈ Ring)
8825ad3antrrr 728 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (mulGrp‘𝐴) ∈ Mnd)
89 simpr 485 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝑧 ∈ ℕ0)
9028adantr 481 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝑀𝐵)
9121, 22, 88, 89, 90mulgnn0cld 18897 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝑧 𝑀) ∈ 𝐵)
9287, 91jca 512 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝐴 ∈ Ring ∧ (𝑧 𝑀) ∈ 𝐵))
9392adantr 481 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → (𝐴 ∈ Ring ∧ (𝑧 𝑀) ∈ 𝐵))
947, 49, 8ringrz 20012 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ Ring ∧ (𝑧 𝑀) ∈ 𝐵) → ((𝑧 𝑀)(.r𝐴)(0g𝐴)) = (0g𝐴))
9593, 94syl 17 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → ((𝑧 𝑀)(.r𝐴)(0g𝐴)) = (0g𝐴))
9677, 86, 953eqtrd 2780 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → 𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (0g𝐴))
9796ex 413 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → ((𝐺𝑧) = (0g𝑌) → 𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (0g𝐴)))
9897adantlr 713 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑤 ∈ ℕ0) ∧ 𝑧 ∈ ℕ0) → ((𝐺𝑧) = (0g𝑌) → 𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (0g𝐴)))
9998imim2d 57 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑤 ∈ ℕ0) ∧ 𝑧 ∈ ℕ0) → ((𝑤 < 𝑧 → (𝐺𝑧) = (0g𝑌)) → (𝑤 < 𝑧𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (0g𝐴))))
10099ralimdva 3164 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑤 ∈ ℕ0) → (∀𝑧 ∈ ℕ0 (𝑤 < 𝑧 → (𝐺𝑧) = (0g𝑌)) → ∀𝑧 ∈ ℕ0 (𝑤 < 𝑧𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (0g𝐴))))
101100reximdva 3165 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (∃𝑤 ∈ ℕ0𝑧 ∈ ℕ0 (𝑤 < 𝑧 → (𝐺𝑧) = (0g𝑌)) → ∃𝑤 ∈ ℕ0𝑧 ∈ ℕ0 (𝑤 < 𝑧𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (0g𝐴))))
10264, 101syld 47 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝐺 finSupp (0g𝑌) → ∃𝑤 ∈ ℕ0𝑧 ∈ ℕ0 (𝑤 < 𝑧𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (0g𝐴))))
10356, 102mpd 15 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → ∃𝑤 ∈ ℕ0𝑧 ∈ ℕ0 (𝑤 < 𝑧𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (0g𝐴)))
10453, 54, 103mptnn0fsupp 13902 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))) finSupp (0g𝐴))
1057, 8, 15, 17, 52, 104gsumcl 19692 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))) ∈ 𝐵)
10632, 9, 7, 43m2cpminvid 22102 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))) ∈ 𝐵) → (𝑈‘(𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))))
1073, 6, 105, 106syl3anc 1371 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝑈‘(𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))))
10838, 39pmatring 22041 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
1092, 5, 108syl2anc 584 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
110 ringmnd 19974 . . . . . . . . 9 (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
111109, 110syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Mnd)
112111ad2antrr 724 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑌 ∈ Mnd)
113 chcoeffeq.w . . . . . . . . . 10 𝑊 = (Base‘𝑌)
11443, 9, 7, 38, 39, 113mat2pmatghm 22079 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑌))
1153, 6, 114syl2anc 584 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑇 ∈ (𝐴 GrpHom 𝑌))
116 ghmmhm 19018 . . . . . . . 8 (𝑇 ∈ (𝐴 GrpHom 𝑌) → 𝑇 ∈ (𝐴 MndHom 𝑌))
117115, 116syl 17 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑇 ∈ (𝐴 MndHom 𝑌))
11823ad3antrrr 728 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
1194, 33sylan2 593 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
1201193adant3 1132 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
121120ad3antrrr 728 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
122121, 47ffvelcdmd 7036 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺𝑛)) ∈ 𝐵)
123118, 30, 122, 50syl3anc 1371 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ 𝐵)
1247, 8, 15, 112, 17, 117, 123, 104gsummptmhm 19717 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝑌 Σg (𝑛 ∈ ℕ0 ↦ (𝑇‘((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))))) = (𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))))))
12543, 9, 7, 38, 39, 113mat2pmatrhm 22083 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑌))
1261253adant3 1132 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑇 ∈ (𝐴 RingHom 𝑌))
127126ad3antrrr 728 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑇 ∈ (𝐴 RingHom 𝑌))
1287, 49, 40rhmmul 20159 . . . . . . . . . 10 ((𝑇 ∈ (𝐴 RingHom 𝑌) ∧ (𝑛 𝑀) ∈ 𝐵 ∧ (𝑈‘(𝐺𝑛)) ∈ 𝐵) → (𝑇‘((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))) = ((𝑇‘(𝑛 𝑀)) × (𝑇‘(𝑈‘(𝐺𝑛)))))
129127, 30, 122, 128syl3anc 1371 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))) = ((𝑇‘(𝑛 𝑀)) × (𝑇‘(𝑈‘(𝐺𝑛)))))
13043, 9, 7, 38, 39, 113mat2pmatmhm 22082 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
1311303adant3 1132 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
132131ad3antrrr 728 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
133 cayhamlem.e2 . . . . . . . . . . . 12 𝐸 = (.g‘(mulGrp‘𝑌))
13421, 22, 133mhmmulg 18917 . . . . . . . . . . 11 ((𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)) ∧ 𝑛 ∈ ℕ0𝑀𝐵) → (𝑇‘(𝑛 𝑀)) = (𝑛𝐸(𝑇𝑀)))
135132, 27, 29, 134syl3anc 1371 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑛 𝑀)) = (𝑛𝐸(𝑇𝑀)))
1362ad3antrrr 728 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
1375ad3antrrr 728 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
13831, 32, 43m2cpminvid2 22104 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐺𝑛) ∈ (𝑁 ConstPolyMat 𝑅)) → (𝑇‘(𝑈‘(𝐺𝑛))) = (𝐺𝑛))
139136, 137, 47, 138syl3anc 1371 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑈‘(𝐺𝑛))) = (𝐺𝑛))
140135, 139oveq12d 7375 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑇‘(𝑛 𝑀)) × (𝑇‘(𝑈‘(𝐺𝑛)))) = ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))
141129, 140eqtrd 2776 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))) = ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))
142141mpteq2dva 5205 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ (𝑇‘((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))) = (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))
143142oveq2d 7373 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝑌 Σg (𝑛 ∈ ℕ0 ↦ (𝑇‘((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))))) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))))
144124, 143eqtr3d 2778 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))))) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))))
145144fveq2d 6846 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝑈‘(𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))))
146107, 145eqtr3d 2778 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))))
1471, 146sylan9eqr 2798 . 2 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑛) (𝑛 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑛) (𝑛 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))))
148 chcoeffeq.c . . 3 𝐶 = (𝑁 CharPlyMat 𝑅)
149 chcoeffeq.k . . 3 𝐾 = (𝐶𝑀)
150 chcoeffeq.1 . . 3 1 = (1r𝐴)
151 chcoeffeq.m . . 3 = ( ·𝑠𝐴)
1529, 7, 38, 39, 40, 41, 42, 43, 148, 149, 44, 113, 150, 151, 32, 22, 49cayhamlem3 22236 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑛) (𝑛 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))))
153147, 152reximddv2 3206 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑛) (𝑛 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  csb 3855  ifcif 4486   class class class wbr 5105  cmpt 5188  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  Fincfn 8883   finSupp cfsupp 9305  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cmin 11385  cn 12153  0cn0 12413  ...cfz 13424  Basecbs 17083  .rcmulr 17134   ·𝑠 cvsca 17137  0gc0g 17321   Σg cgsu 17322  Mndcmnd 18556   MndHom cmhm 18599  -gcsg 18750  .gcmg 18872   GrpHom cghm 19005  CMndccmn 19562  mulGrpcmgp 19896  1rcur 19913  Ringcrg 19964  CRingccrg 19965   RingHom crh 20143  Poly1cpl1 21548  coe1cco1 21549   Mat cmat 21754   ConstPolyMat ccpmat 22052   matToPolyMat cmat2pmat 22053   cPolyMatToMat ccpmat2mat 22054   CharPlyMat cchpmat 22175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-xor 1510  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-tpos 8157  df-cur 8198  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-dec 12619  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-splice 14638  df-reverse 14647  df-s2 14737  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-efmnd 18679  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-gim 19049  df-cntz 19097  df-oppg 19124  df-symg 19149  df-pmtr 19224  df-psgn 19273  df-evpm 19274  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-srg 19918  df-ring 19966  df-cring 19967  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-rnghom 20146  df-drng 20187  df-subrg 20220  df-lmod 20324  df-lss 20393  df-sra 20633  df-rgmod 20634  df-cnfld 20797  df-zring 20870  df-zrh 20904  df-dsmm 21138  df-frlm 21153  df-assa 21259  df-ascl 21261  df-psr 21311  df-mvr 21312  df-mpl 21313  df-opsr 21315  df-psr1 21551  df-vr1 21552  df-ply1 21553  df-coe1 21554  df-mamu 21733  df-mat 21755  df-mdet 21934  df-madu 21983  df-cpmat 22055  df-mat2pmat 22056  df-cpmat2mat 22057  df-decpmat 22112  df-pm2mp 22142  df-chpmat 22176
This theorem is referenced by:  cayleyhamilton0  22238  cayleyhamiltonALT  22240
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