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Theorem cayhamlem4 21424
Description: Lemma for cayleyhamilton 21426. (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 1128 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑁 ∈ Fin)
32ad2antrr 722 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑁 ∈ Fin)
4 crngring 19237 . . . . . . 7 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
543ad2ant2 1126 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
65ad2antrr 722 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑅 ∈ Ring)
7 chcoeffeq.b . . . . . 6 𝐵 = (Base‘𝐴)
8 eqid 2818 . . . . . 6 (0g𝐴) = (0g𝐴)
9 chcoeffeq.a . . . . . . . . . . 11 𝐴 = (𝑁 Mat 𝑅)
109matring 20980 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
114, 10sylan2 592 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
12 ringcmn 19260 . . . . . . . . 9 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
1311, 12syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ CMnd)
14133adant3 1124 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ CMnd)
1514ad2antrr 722 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝐴 ∈ CMnd)
16 nn0ex 11891 . . . . . . 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)
202, 5, 10syl2anc 584 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ Ring)
21 eqid 2818 . . . . . . . . . . . 12 (mulGrp‘𝐴) = (mulGrp‘𝐴)
2221ringmgp 19232 . . . . . . . . . . 11 (𝐴 ∈ Ring → (mulGrp‘𝐴) ∈ Mnd)
2320, 22syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (mulGrp‘𝐴) ∈ Mnd)
2423ad3antrrr 726 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (mulGrp‘𝐴) ∈ Mnd)
25 simpr 485 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
26 simpll3 1206 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑀𝐵)
2726adantr 481 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑀𝐵)
2821, 7mgpbas 19174 . . . . . . . . . 10 𝐵 = (Base‘(mulGrp‘𝐴))
29 cayhamlem.e1 . . . . . . . . . 10 = (.g‘(mulGrp‘𝐴))
3028, 29mulgnn0cl 18182 . . . . . . . . 9 (((mulGrp‘𝐴) ∈ Mnd ∧ 𝑛 ∈ ℕ0𝑀𝐵) → (𝑛 𝑀) ∈ 𝐵)
3124, 25, 27, 30syl3anc 1363 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑀) ∈ 𝐵)
32 eqid 2818 . . . . . . . . . . . 12 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
33 chcoeffeq.u . . . . . . . . . . . 12 𝑈 = (𝑁 cPolyMatToMat 𝑅)
349, 7, 32, 33cpm2mf 21288 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
352, 5, 34syl2anc 584 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
3635ad3antrrr 726 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
37 simplr 765 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑠 ∈ ℕ)
38 simpr 485 . . . . . . . . . . 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 21391 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
473, 6, 26, 37, 38, 46syl32anc 1370 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
4847ffvelrnda 6843 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (𝑁 ConstPolyMat 𝑅))
4936, 48ffvelrnd 6844 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺𝑛)) ∈ 𝐵)
50 eqid 2818 . . . . . . . . 9 (.r𝐴) = (.r𝐴)
517, 50ringcl 19240 . . . . . . . 8 ((𝐴 ∈ Ring ∧ (𝑛 𝑀) ∈ 𝐵 ∧ (𝑈‘(𝐺𝑛)) ∈ 𝐵) → ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ 𝐵)
5219, 31, 49, 51syl3anc 1363 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ 𝐵)
5352fmpttd 6871 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))):ℕ0𝐵)
54 fvexd 6678 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (0g𝐴) ∈ V)
55 ovexd 7180 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ V)
569, 7, 39, 40, 41, 42, 43, 44, 45chfacffsupp 21392 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝐺 finSupp (0g𝑌))
5756anassrs 468 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝐺 finSupp (0g𝑌))
58 ovex 7178 . . . . . . . . . . . . 13 (𝑁 ConstPolyMat 𝑅) ∈ V
5958, 16pm3.2i 471 . . . . . . . . . . . 12 ((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ0 ∈ V)
60 elmapg 8408 . . . . . . . . . . . 12 (((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ0 ∈ V) → (𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m0) ↔ 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)))
6159, 60mp1i 13 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m0) ↔ 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)))
6247, 61mpbird 258 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m0))
63 fvex 6676 . . . . . . . . . 10 (0g𝑌) ∈ V
64 fsuppmapnn0ub 13351 . . . . . . . . . 10 ((𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m0) ∧ (0g𝑌) ∈ V) → (𝐺 finSupp (0g𝑌) → ∃𝑤 ∈ ℕ0𝑧 ∈ ℕ0 (𝑤 < 𝑧 → (𝐺𝑧) = (0g𝑌))))
6562, 63, 64sylancl 586 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝐺 finSupp (0g𝑌) → ∃𝑤 ∈ ℕ0𝑧 ∈ ℕ0 (𝑤 < 𝑧 → (𝐺𝑧) = (0g𝑌))))
66 csbov12g 7189 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (𝑧 / 𝑛(𝑛 𝑀)(.r𝐴)𝑧 / 𝑛(𝑈‘(𝐺𝑛))))
67 csbov1g 7190 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑛 𝑀) = (𝑧 / 𝑛𝑛 𝑀))
68 csbvarg 4380 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℕ0𝑧 / 𝑛𝑛 = 𝑧)
6968oveq1d 7160 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0 → (𝑧 / 𝑛𝑛 𝑀) = (𝑧 𝑀))
7067, 69eqtrd 2853 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑛 𝑀) = (𝑧 𝑀))
71 csbfv2g 6707 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑈‘(𝐺𝑛)) = (𝑈𝑧 / 𝑛(𝐺𝑛)))
72 csbfv 6708 . . . . . . . . . . . . . . . . . . . . 21 𝑧 / 𝑛(𝐺𝑛) = (𝐺𝑧)
7372a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝐺𝑛) = (𝐺𝑧))
7473fveq2d 6667 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0 → (𝑈𝑧 / 𝑛(𝐺𝑛)) = (𝑈‘(𝐺𝑧)))
7571, 74eqtrd 2853 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑈‘(𝐺𝑛)) = (𝑈‘(𝐺𝑧)))
7670, 75oveq12d 7163 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0 → (𝑧 / 𝑛(𝑛 𝑀)(.r𝐴)𝑧 / 𝑛(𝑈‘(𝐺𝑛))) = ((𝑧 𝑀)(.r𝐴)(𝑈‘(𝐺𝑧))))
7766, 76eqtrd 2853 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ℕ0𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = ((𝑧 𝑀)(.r𝐴)(𝑈‘(𝐺𝑧))))
7877ad2antlr 723 . . . . . . . . . . . . . . 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))))
79 fveq2 6663 . . . . . . . . . . . . . . . . 17 ((𝐺𝑧) = (0g𝑌) → (𝑈‘(𝐺𝑧)) = (𝑈‘(0g𝑌)))
802, 5jca 512 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
8180adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
82 eqid 2818 . . . . . . . . . . . . . . . . . . . 20 (0g𝑌) = (0g𝑌)
839, 33, 39, 40, 8, 82m2cpminv0 21297 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑈‘(0g𝑌)) = (0g𝐴))
8481, 83syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) → (𝑈‘(0g𝑌)) = (0g𝐴))
8584ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝑈‘(0g𝑌)) = (0g𝐴))
8679, 85sylan9eqr 2875 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → (𝑈‘(𝐺𝑧)) = (0g𝐴))
8786oveq2d 7161 . . . . . . . . . . . . . . 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)))
8818adantr 481 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝐴 ∈ Ring)
8923ad3antrrr 726 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (mulGrp‘𝐴) ∈ Mnd)
90 simpr 485 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝑧 ∈ ℕ0)
9126adantr 481 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝑀𝐵)
9228, 29mulgnn0cl 18182 . . . . . . . . . . . . . . . . . . 19 (((mulGrp‘𝐴) ∈ Mnd ∧ 𝑧 ∈ ℕ0𝑀𝐵) → (𝑧 𝑀) ∈ 𝐵)
9389, 90, 91, 92syl3anc 1363 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝑧 𝑀) ∈ 𝐵)
9488, 93jca 512 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝐴 ∈ Ring ∧ (𝑧 𝑀) ∈ 𝐵))
9594adantr 481 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → (𝐴 ∈ Ring ∧ (𝑧 𝑀) ∈ 𝐵))
967, 50, 8ringrz 19267 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ Ring ∧ (𝑧 𝑀) ∈ 𝐵) → ((𝑧 𝑀)(.r𝐴)(0g𝐴)) = (0g𝐴))
9795, 96syl 17 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → ((𝑧 𝑀)(.r𝐴)(0g𝐴)) = (0g𝐴))
9878, 87, 973eqtrd 2857 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → 𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (0g𝐴))
9998ex 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)))
10099adantlr 711 . . . . . . . . . . . 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)))
101100imim2d 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))))
102101ralimdva 3174 . . . . . . . . . 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))))
103102reximdva 3271 . . . . . . . . 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))))
10465, 103syld 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))))
10557, 104mpd 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)))
10654, 55, 105mptnn0fsupp 13353 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))) finSupp (0g𝐴))
1077, 8, 15, 17, 53, 106gsumcl 18964 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))))) ∈ 𝐵)
10833, 9, 7, 44m2cpminvid 21289 . . . . 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))))))
1093, 6, 107, 108syl3anc 1363 . . . 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))))))
11039, 40pmatring 21229 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
1112, 5, 110syl2anc 584 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
112 ringmnd 19235 . . . . . . . . 9 (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
113111, 112syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Mnd)
114113ad2antrr 722 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑌 ∈ Mnd)
115 chcoeffeq.w . . . . . . . . . 10 𝑊 = (Base‘𝑌)
11644, 9, 7, 39, 40, 115mat2pmatghm 21266 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑌))
1173, 6, 116syl2anc 584 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑇 ∈ (𝐴 GrpHom 𝑌))
118 ghmmhm 18306 . . . . . . . 8 (𝑇 ∈ (𝐴 GrpHom 𝑌) → 𝑇 ∈ (𝐴 MndHom 𝑌))
119117, 118syl 17 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑇 ∈ (𝐴 MndHom 𝑌))
12020ad3antrrr 726 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
1214, 34sylan2 592 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
1221213adant3 1124 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
123122ad3antrrr 726 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
124123, 48ffvelrnd 6844 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺𝑛)) ∈ 𝐵)
125120, 31, 124, 51syl3anc 1363 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ 𝐵)
1267, 8, 15, 114, 17, 119, 125, 106gsummptmhm 18989 . . . . . 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)))))))
12744, 9, 7, 39, 40, 115mat2pmatrhm 21270 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑌))
1281273adant3 1124 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑇 ∈ (𝐴 RingHom 𝑌))
129128ad3antrrr 726 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑇 ∈ (𝐴 RingHom 𝑌))
1307, 50, 41rhmmul 19408 . . . . . . . . . 10 ((𝑇 ∈ (𝐴 RingHom 𝑌) ∧ (𝑛 𝑀) ∈ 𝐵 ∧ (𝑈‘(𝐺𝑛)) ∈ 𝐵) → (𝑇‘((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))) = ((𝑇‘(𝑛 𝑀)) × (𝑇‘(𝑈‘(𝐺𝑛)))))
131129, 31, 124, 130syl3anc 1363 . . . . . . . . 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)))))
13244, 9, 7, 39, 40, 115mat2pmatmhm 21269 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
1331323adant3 1124 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
134133ad3antrrr 726 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
135 cayhamlem.e2 . . . . . . . . . . . 12 𝐸 = (.g‘(mulGrp‘𝑌))
13628, 29, 135mhmmulg 18206 . . . . . . . . . . 11 ((𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)) ∧ 𝑛 ∈ ℕ0𝑀𝐵) → (𝑇‘(𝑛 𝑀)) = (𝑛𝐸(𝑇𝑀)))
137134, 25, 27, 136syl3anc 1363 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑛 𝑀)) = (𝑛𝐸(𝑇𝑀)))
1382ad3antrrr 726 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
1395ad3antrrr 726 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
14032, 33, 44m2cpminvid2 21291 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐺𝑛) ∈ (𝑁 ConstPolyMat 𝑅)) → (𝑇‘(𝑈‘(𝐺𝑛))) = (𝐺𝑛))
141138, 139, 48, 140syl3anc 1363 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑈‘(𝐺𝑛))) = (𝐺𝑛))
142137, 141oveq12d 7163 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑇‘(𝑛 𝑀)) × (𝑇‘(𝑈‘(𝐺𝑛)))) = ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))
143131, 142eqtrd 2853 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))) = ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))
144143mpteq2dva 5152 . . . . . . 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))))
145144oveq2d 7161 . . . . . 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)))))
146126, 145eqtr3d 2855 . . . . 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)))))
147146fveq2d 6667 . . . 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))))))
148109, 147eqtr3d 2855 . . 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))))))
1491, 148sylan9eqr 2875 . 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))))))
150 chcoeffeq.c . . 3 𝐶 = (𝑁 CharPlyMat 𝑅)
151 chcoeffeq.k . . 3 𝐾 = (𝐶𝑀)
152 chcoeffeq.1 . . 3 1 = (1r𝐴)
153 chcoeffeq.m . . 3 = ( ·𝑠𝐴)
1549, 7, 39, 40, 41, 42, 43, 44, 150, 151, 45, 115, 152, 153, 33, 29, 50cayhamlem3 21423 . 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))))))
155149, 154reximddv2 3275 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  Vcvv 3492  csb 3880  ifcif 4463   class class class wbr 5057  cmpt 5137  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395  Fincfn 8497   finSupp cfsupp 8821  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663  cmin 10858  cn 11626  0cn0 11885  ...cfz 12880  Basecbs 16471  .rcmulr 16554   ·𝑠 cvsca 16557  0gc0g 16701   Σg cgsu 16702  Mndcmnd 17899   MndHom cmhm 17942  -gcsg 18043  .gcmg 18162   GrpHom cghm 18293  CMndccmn 18835  mulGrpcmgp 19168  1rcur 19180  Ringcrg 19226  CRingccrg 19227   RingHom crh 19393  Poly1cpl1 20273  coe1cco1 20274   Mat cmat 20944   ConstPolyMat ccpmat 21239   matToPolyMat cmat2pmat 21240   cPolyMatToMat ccpmat2mat 21241   CharPlyMat cchpmat 21362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-addf 10604  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-xor 1496  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-ot 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-ofr 7399  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-tpos 7881  df-cur 7922  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-xnn0 11956  df-z 11970  df-dec 12087  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-word 13850  df-lsw 13903  df-concat 13911  df-s1 13938  df-substr 13991  df-pfx 14021  df-splice 14100  df-reverse 14109  df-s2 14198  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-starv 16568  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-hom 16577  df-cco 16578  df-0g 16703  df-gsum 16704  df-prds 16709  df-pws 16711  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-submnd 17945  df-grp 18044  df-minusg 18045  df-sbg 18046  df-mulg 18163  df-subg 18214  df-ghm 18294  df-gim 18337  df-cntz 18385  df-oppg 18412  df-symg 18434  df-pmtr 18499  df-psgn 18548  df-evpm 18549  df-cmn 18837  df-abl 18838  df-mgp 19169  df-ur 19181  df-srg 19185  df-ring 19228  df-cring 19229  df-oppr 19302  df-dvdsr 19320  df-unit 19321  df-invr 19351  df-dvr 19362  df-rnghom 19396  df-drng 19433  df-subrg 19462  df-lmod 19565  df-lss 19633  df-sra 19873  df-rgmod 19874  df-assa 20013  df-ascl 20015  df-psr 20064  df-mvr 20065  df-mpl 20066  df-opsr 20068  df-psr1 20276  df-vr1 20277  df-ply1 20278  df-coe1 20279  df-cnfld 20474  df-zring 20546  df-zrh 20579  df-dsmm 20804  df-frlm 20819  df-mamu 20923  df-mat 20945  df-mdet 21122  df-madu 21171  df-cpmat 21242  df-mat2pmat 21243  df-cpmat2mat 21244  df-decpmat 21299  df-pm2mp 21329  df-chpmat 21363
This theorem is referenced by:  cayleyhamilton0  21425  cayleyhamiltonALT  21427
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