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Theorem cayhamlem4 20512
Description: Lemma for cayleyhamilton 20514. (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 1054 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑁 ∈ Fin)
32ad2antrr 758 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝑁 ∈ Fin)
4 crngring 18381 . . . . . . 7 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
543ad2ant2 1076 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
65ad2antrr 758 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝑅 ∈ Ring)
7 chcoeffeq.b . . . . . 6 𝐵 = (Base‘𝐴)
8 eqid 2610 . . . . . 6 (0g𝐴) = (0g𝐴)
9 chcoeffeq.a . . . . . . . . . . 11 𝐴 = (𝑁 Mat 𝑅)
109matring 20068 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
114, 10sylan2 490 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
12 ringcmn 18404 . . . . . . . . 9 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
1311, 12syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ CMnd)
14133adant3 1074 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ CMnd)
1514ad2antrr 758 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝐴 ∈ CMnd)
16 nn0ex 11175 . . . . . . 7 0 ∈ V
1716a1i 11 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → ℕ0 ∈ V)
183, 6, 10syl2anc 691 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝐴 ∈ Ring)
1918adantr 480 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
202, 5, 10syl2anc 691 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ Ring)
21 eqid 2610 . . . . . . . . . . . 12 (mulGrp‘𝐴) = (mulGrp‘𝐴)
2221ringmgp 18376 . . . . . . . . . . 11 (𝐴 ∈ Ring → (mulGrp‘𝐴) ∈ Mnd)
2320, 22syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (mulGrp‘𝐴) ∈ Mnd)
2423ad3antrrr 762 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (mulGrp‘𝐴) ∈ Mnd)
25 simpr 476 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
26 simpll3 1095 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝑀𝐵)
2726adantr 480 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑀𝐵)
2821, 7mgpbas 18318 . . . . . . . . . 10 𝐵 = (Base‘(mulGrp‘𝐴))
29 cayhamlem.e1 . . . . . . . . . 10 = (.g‘(mulGrp‘𝐴))
3028, 29mulgnn0cl 17381 . . . . . . . . 9 (((mulGrp‘𝐴) ∈ Mnd ∧ 𝑛 ∈ ℕ0𝑀𝐵) → (𝑛 𝑀) ∈ 𝐵)
3124, 25, 27, 30syl3anc 1318 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑀) ∈ 𝐵)
32 eqid 2610 . . . . . . . . . . . 12 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
33 chcoeffeq.u . . . . . . . . . . . 12 𝑈 = (𝑁 cPolyMatToMat 𝑅)
349, 7, 32, 33cpm2mf 20376 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
352, 5, 34syl2anc 691 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
3635ad3antrrr 762 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
37 simplr 788 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝑠 ∈ ℕ)
38 simpr 476 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝑏 ∈ (𝐵𝑚 (0...𝑠)))
39 chcoeffeq.p . . . . . . . . . . . 12 𝑃 = (Poly1𝑅)
40 chcoeffeq.y . . . . . . . . . . . 12 𝑌 = (𝑁 Mat 𝑃)
41 chcoeffeq.r . . . . . . . . . . . 12 × = (.r𝑌)
42 chcoeffeq.s . . . . . . . . . . . 12 = (-g𝑌)
43 chcoeffeq.0 . . . . . . . . . . . 12 0 = (0g𝑌)
44 chcoeffeq.t . . . . . . . . . . . 12 𝑇 = (𝑁 matToPolyMat 𝑅)
45 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))))))))
469, 7, 39, 40, 41, 42, 43, 44, 45, 32chfacfisfcpmat 20479 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
473, 6, 26, 37, 38, 46syl32anc 1326 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
4847ffvelrnda 6267 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (𝑁 ConstPolyMat 𝑅))
4936, 48ffvelrnd 6268 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺𝑛)) ∈ 𝐵)
50 eqid 2610 . . . . . . . . 9 (.r𝐴) = (.r𝐴)
517, 50ringcl 18384 . . . . . . . 8 ((𝐴 ∈ Ring ∧ (𝑛 𝑀) ∈ 𝐵 ∧ (𝑈‘(𝐺𝑛)) ∈ 𝐵) → ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ 𝐵)
5219, 31, 49, 51syl3anc 1318 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ 𝐵)
53 eqid 2610 . . . . . . 7 (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))) = (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))
5452, 53fmptd 6292 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))):ℕ0𝐵)
55 fvex 6113 . . . . . . . 8 (0g𝐴) ∈ V
5655a1i 11 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (0g𝐴) ∈ V)
57 ovex 6577 . . . . . . . 8 ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ V
5857a1i 11 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ V)
599, 7, 39, 40, 41, 42, 43, 44, 45chfacffsupp 20480 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺 finSupp (0g𝑌))
6059anassrs 678 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝐺 finSupp (0g𝑌))
61 ovex 6577 . . . . . . . . . . . . 13 (𝑁 ConstPolyMat 𝑅) ∈ V
6261, 16pm3.2i 470 . . . . . . . . . . . 12 ((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ0 ∈ V)
63 elmapg 7757 . . . . . . . . . . . 12 (((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ0 ∈ V) → (𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑𝑚0) ↔ 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)))
6462, 63mp1i 13 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑𝑚0) ↔ 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)))
6547, 64mpbird 246 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑𝑚0))
66 fvex 6113 . . . . . . . . . 10 (0g𝑌) ∈ V
67 fsuppmapnn0ub 12657 . . . . . . . . . 10 ((𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑𝑚0) ∧ (0g𝑌) ∈ V) → (𝐺 finSupp (0g𝑌) → ∃𝑤 ∈ ℕ0𝑧 ∈ ℕ0 (𝑤 < 𝑧 → (𝐺𝑧) = (0g𝑌))))
6865, 66, 67sylancl 693 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝐺 finSupp (0g𝑌) → ∃𝑤 ∈ ℕ0𝑧 ∈ ℕ0 (𝑤 < 𝑧 → (𝐺𝑧) = (0g𝑌))))
69 csbov12g 6587 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (𝑧 / 𝑛(𝑛 𝑀)(.r𝐴)𝑧 / 𝑛(𝑈‘(𝐺𝑛))))
70 csbov1g 6588 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑛 𝑀) = (𝑧 / 𝑛𝑛 𝑀))
71 csbvarg 3955 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℕ0𝑧 / 𝑛𝑛 = 𝑧)
7271oveq1d 6564 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0 → (𝑧 / 𝑛𝑛 𝑀) = (𝑧 𝑀))
7370, 72eqtrd 2644 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑛 𝑀) = (𝑧 𝑀))
74 csbfv2g 6142 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑈‘(𝐺𝑛)) = (𝑈𝑧 / 𝑛(𝐺𝑛)))
75 csbfv 6143 . . . . . . . . . . . . . . . . . . . . 21 𝑧 / 𝑛(𝐺𝑛) = (𝐺𝑧)
7675a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝐺𝑛) = (𝐺𝑧))
7776fveq2d 6107 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0 → (𝑈𝑧 / 𝑛(𝐺𝑛)) = (𝑈‘(𝐺𝑧)))
7874, 77eqtrd 2644 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑈‘(𝐺𝑛)) = (𝑈‘(𝐺𝑧)))
7973, 78oveq12d 6567 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0 → (𝑧 / 𝑛(𝑛 𝑀)(.r𝐴)𝑧 / 𝑛(𝑈‘(𝐺𝑛))) = ((𝑧 𝑀)(.r𝐴)(𝑈‘(𝐺𝑧))))
8069, 79eqtrd 2644 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ℕ0𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = ((𝑧 𝑀)(.r𝐴)(𝑈‘(𝐺𝑧))))
8180ad2antlr 759 . . . . . . . . . . . . . . 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))))
82 fveq2 6103 . . . . . . . . . . . . . . . . 17 ((𝐺𝑧) = (0g𝑌) → (𝑈‘(𝐺𝑧)) = (𝑈‘(0g𝑌)))
832, 5jca 553 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
8483adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
85 eqid 2610 . . . . . . . . . . . . . . . . . . . 20 (0g𝑌) = (0g𝑌)
869, 33, 39, 40, 8, 85m2cpminv0 20385 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑈‘(0g𝑌)) = (0g𝐴))
8784, 86syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) → (𝑈‘(0g𝑌)) = (0g𝐴))
8887ad2antrr 758 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝑈‘(0g𝑌)) = (0g𝐴))
8982, 88sylan9eqr 2666 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → (𝑈‘(𝐺𝑧)) = (0g𝐴))
9089oveq2d 6565 . . . . . . . . . . . . . . 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)))
9118adantr 480 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝐴 ∈ Ring)
9223ad3antrrr 762 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (mulGrp‘𝐴) ∈ Mnd)
93 simpr 476 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝑧 ∈ ℕ0)
9426adantr 480 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝑀𝐵)
9528, 29mulgnn0cl 17381 . . . . . . . . . . . . . . . . . . 19 (((mulGrp‘𝐴) ∈ Mnd ∧ 𝑧 ∈ ℕ0𝑀𝐵) → (𝑧 𝑀) ∈ 𝐵)
9692, 93, 94, 95syl3anc 1318 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝑧 𝑀) ∈ 𝐵)
9791, 96jca 553 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝐴 ∈ Ring ∧ (𝑧 𝑀) ∈ 𝐵))
9897adantr 480 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → (𝐴 ∈ Ring ∧ (𝑧 𝑀) ∈ 𝐵))
997, 50, 8ringrz 18411 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ Ring ∧ (𝑧 𝑀) ∈ 𝐵) → ((𝑧 𝑀)(.r𝐴)(0g𝐴)) = (0g𝐴))
10098, 99syl 17 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → ((𝑧 𝑀)(.r𝐴)(0g𝐴)) = (0g𝐴))
10181, 90, 1003eqtrd 2648 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → 𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (0g𝐴))
102101ex 449 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → ((𝐺𝑧) = (0g𝑌) → 𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (0g𝐴)))
103102adantlr 747 . . . . . . . . . . . 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)))
104103imim2d 55 . . . . . . . . . . 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))))
105104ralimdva 2945 . . . . . . . . . 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))))
106105reximdva 3000 . . . . . . . . 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))))
10768, 106syld 46 . . . . . . . 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))))
10860, 107mpd 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)))
10956, 58, 108mptnn0fsupp 12659 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))) finSupp (0g𝐴))
1107, 8, 15, 17, 54, 109gsumcl 18139 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))) ∈ 𝐵)
11133, 9, 7, 44m2cpminvid 20377 . . . . 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))))))
1123, 6, 110, 111syl3anc 1318 . . . 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))))))
11339, 40pmatring 20317 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
1142, 5, 113syl2anc 691 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
115 ringmnd 18379 . . . . . . . . 9 (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
116114, 115syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Mnd)
117116ad2antrr 758 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝑌 ∈ Mnd)
118 chcoeffeq.w . . . . . . . . . 10 𝑊 = (Base‘𝑌)
11944, 9, 7, 39, 40, 118mat2pmatghm 20354 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑌))
1203, 6, 119syl2anc 691 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝑇 ∈ (𝐴 GrpHom 𝑌))
121 ghmmhm 17493 . . . . . . . 8 (𝑇 ∈ (𝐴 GrpHom 𝑌) → 𝑇 ∈ (𝐴 MndHom 𝑌))
122120, 121syl 17 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝑇 ∈ (𝐴 MndHom 𝑌))
12320ad3antrrr 762 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
1244, 34sylan2 490 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
1251243adant3 1074 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
126125ad3antrrr 762 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
127126, 48ffvelrnd 6268 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺𝑛)) ∈ 𝐵)
128123, 31, 127, 51syl3anc 1318 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ 𝐵)
1297, 8, 15, 117, 17, 122, 128, 109gsummptmhm 18163 . . . . . 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)))))))
13044, 9, 7, 39, 40, 118mat2pmatrhm 20358 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑌))
1311303adant3 1074 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑇 ∈ (𝐴 RingHom 𝑌))
132131ad3antrrr 762 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑇 ∈ (𝐴 RingHom 𝑌))
1337, 50, 41rhmmul 18550 . . . . . . . . . 10 ((𝑇 ∈ (𝐴 RingHom 𝑌) ∧ (𝑛 𝑀) ∈ 𝐵 ∧ (𝑈‘(𝐺𝑛)) ∈ 𝐵) → (𝑇‘((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))) = ((𝑇‘(𝑛 𝑀)) × (𝑇‘(𝑈‘(𝐺𝑛)))))
134132, 31, 127, 133syl3anc 1318 . . . . . . . . 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)))))
13544, 9, 7, 39, 40, 118mat2pmatmhm 20357 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
1361353adant3 1074 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
137136ad3antrrr 762 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
138 cayhamlem.e2 . . . . . . . . . . . 12 𝐸 = (.g‘(mulGrp‘𝑌))
13928, 29, 138mhmmulg 17406 . . . . . . . . . . 11 ((𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)) ∧ 𝑛 ∈ ℕ0𝑀𝐵) → (𝑇‘(𝑛 𝑀)) = (𝑛𝐸(𝑇𝑀)))
140137, 25, 27, 139syl3anc 1318 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑛 𝑀)) = (𝑛𝐸(𝑇𝑀)))
1412ad3antrrr 762 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
1425ad3antrrr 762 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
14332, 33, 44m2cpminvid2 20379 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐺𝑛) ∈ (𝑁 ConstPolyMat 𝑅)) → (𝑇‘(𝑈‘(𝐺𝑛))) = (𝐺𝑛))
144141, 142, 48, 143syl3anc 1318 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑈‘(𝐺𝑛))) = (𝐺𝑛))
145140, 144oveq12d 6567 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑇‘(𝑛 𝑀)) × (𝑇‘(𝑈‘(𝐺𝑛)))) = ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))
146134, 145eqtrd 2644 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))) = ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))
147146mpteq2dva 4672 . . . . . . 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))))
148147oveq2d 6565 . . . . . 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)))))
149129, 148eqtr3d 2646 . . . . 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)))))
150149fveq2d 6107 . . . 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))))))
151112, 150eqtr3d 2646 . . 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))))))
1521, 151sylan9eqr 2666 . 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))))))
153 chcoeffeq.c . . 3 𝐶 = (𝑁 CharPlyMat 𝑅)
154 chcoeffeq.k . . 3 𝐾 = (𝐶𝑀)
155 chcoeffeq.1 . . 3 1 = (1r𝐴)
156 chcoeffeq.m . . 3 = ( ·𝑠𝐴)
1579, 7, 39, 40, 41, 42, 43, 44, 153, 154, 45, 118, 155, 156, 33, 29, 50cayhamlem3 20511 . 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))))))
158152, 157reximddv2 3002 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897  Vcvv 3173  csb 3499  ifcif 4036   class class class wbr 4583  cmpt 4643  wf 5800  cfv 5804  (class class class)co 6549  𝑚 cmap 7744  Fincfn 7841   finSupp cfsupp 8158  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953  cmin 10145  cn 10897  0cn0 11169  ...cfz 12197  Basecbs 15695  .rcmulr 15769   ·𝑠 cvsca 15772  0gc0g 15923   Σg cgsu 15924  Mndcmnd 17117   MndHom cmhm 17156  -gcsg 17247  .gcmg 17363   GrpHom cghm 17480  CMndccmn 18016  mulGrpcmgp 18312  1rcur 18324  Ringcrg 18370  CRingccrg 18371   RingHom crh 18535  Poly1cpl1 19368  coe1cco1 19369   Mat cmat 20032   ConstPolyMat ccpmat 20327   matToPolyMat cmat2pmat 20328   cPolyMatToMat ccpmat2mat 20329   CharPlyMat cchpmat 20450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-addf 9894  ax-mulf 9895
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-xor 1457  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-tpos 7239  df-cur 7280  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-reverse 13160  df-s2 13444  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-gim 17524  df-cntz 17573  df-oppg 17599  df-symg 17621  df-pmtr 17685  df-psgn 17734  df-evpm 17735  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-srg 18329  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-rnghom 18538  df-drng 18572  df-subrg 18601  df-lmod 18688  df-lss 18754  df-sra 18993  df-rgmod 18994  df-assa 19133  df-ascl 19135  df-psr 19177  df-mvr 19178  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-vr1 19372  df-ply1 19373  df-coe1 19374  df-cnfld 19568  df-zring 19638  df-zrh 19671  df-dsmm 19895  df-frlm 19910  df-mamu 20009  df-mat 20033  df-mdet 20210  df-madu 20259  df-cpmat 20330  df-mat2pmat 20331  df-cpmat2mat 20332  df-decpmat 20387  df-pm2mp 20417  df-chpmat 20451
This theorem is referenced by:  cayleyhamilton0  20513  cayleyhamiltonALT  20515
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