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Theorem cayhamlem4 21491
Description: Lemma for cayleyhamilton 21493. (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 1133 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑁 ∈ Fin)
32ad2antrr 725 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑁 ∈ Fin)
4 crngring 19300 . . . . . . 7 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
543ad2ant2 1131 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
65ad2antrr 725 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑅 ∈ Ring)
7 chcoeffeq.b . . . . . 6 𝐵 = (Base‘𝐴)
8 eqid 2822 . . . . . 6 (0g𝐴) = (0g𝐴)
9 chcoeffeq.a . . . . . . . . . . 11 𝐴 = (𝑁 Mat 𝑅)
109matring 21046 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
114, 10sylan2 595 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
12 ringcmn 19325 . . . . . . . . 9 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
1311, 12syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ CMnd)
14133adant3 1129 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ CMnd)
1514ad2antrr 725 . . . . . 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 587 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝐴 ∈ Ring)
1918adantr 484 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
202, 5, 10syl2anc 587 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ Ring)
21 eqid 2822 . . . . . . . . . . . 12 (mulGrp‘𝐴) = (mulGrp‘𝐴)
2221ringmgp 19294 . . . . . . . . . . 11 (𝐴 ∈ Ring → (mulGrp‘𝐴) ∈ Mnd)
2320, 22syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (mulGrp‘𝐴) ∈ Mnd)
2423ad3antrrr 729 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (mulGrp‘𝐴) ∈ Mnd)
25 simpr 488 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
26 simpll3 1211 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑀𝐵)
2726adantr 484 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑀𝐵)
2821, 7mgpbas 19236 . . . . . . . . . 10 𝐵 = (Base‘(mulGrp‘𝐴))
29 cayhamlem.e1 . . . . . . . . . 10 = (.g‘(mulGrp‘𝐴))
3028, 29mulgnn0cl 18235 . . . . . . . . 9 (((mulGrp‘𝐴) ∈ Mnd ∧ 𝑛 ∈ ℕ0𝑀𝐵) → (𝑛 𝑀) ∈ 𝐵)
3124, 25, 27, 30syl3anc 1368 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑀) ∈ 𝐵)
32 eqid 2822 . . . . . . . . . . . 12 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
33 chcoeffeq.u . . . . . . . . . . . 12 𝑈 = (𝑁 cPolyMatToMat 𝑅)
349, 7, 32, 33cpm2mf 21355 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
352, 5, 34syl2anc 587 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
3635ad3antrrr 729 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
37 simplr 768 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑠 ∈ ℕ)
38 simpr 488 . . . . . . . . . . 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 21458 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
473, 6, 26, 37, 38, 46syl32anc 1375 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
4847ffvelrnda 6833 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (𝑁 ConstPolyMat 𝑅))
4936, 48ffvelrnd 6834 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺𝑛)) ∈ 𝐵)
50 eqid 2822 . . . . . . . . 9 (.r𝐴) = (.r𝐴)
517, 50ringcl 19305 . . . . . . . 8 ((𝐴 ∈ Ring ∧ (𝑛 𝑀) ∈ 𝐵 ∧ (𝑈‘(𝐺𝑛)) ∈ 𝐵) → ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ 𝐵)
5219, 31, 49, 51syl3anc 1368 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) ∈ 𝐵)
5352fmpttd 6861 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))):ℕ0𝐵)
54 fvexd 6667 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (0g𝐴) ∈ V)
55 ovexd 7175 . . . . . . 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 21459 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝐺 finSupp (0g𝑌))
5756anassrs 471 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝐺 finSupp (0g𝑌))
58 ovex 7173 . . . . . . . . . . . . 13 (𝑁 ConstPolyMat 𝑅) ∈ V
5958, 16pm3.2i 474 . . . . . . . . . . . 12 ((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ0 ∈ V)
60 elmapg 8406 . . . . . . . . . . . 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 260 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m0))
63 fvex 6665 . . . . . . . . . 10 (0g𝑌) ∈ V
64 fsuppmapnn0ub 13358 . . . . . . . . . 10 ((𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m0) ∧ (0g𝑌) ∈ V) → (𝐺 finSupp (0g𝑌) → ∃𝑤 ∈ ℕ0𝑧 ∈ ℕ0 (𝑤 < 𝑧 → (𝐺𝑧) = (0g𝑌))))
6562, 63, 64sylancl 589 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝐺 finSupp (0g𝑌) → ∃𝑤 ∈ ℕ0𝑧 ∈ ℕ0 (𝑤 < 𝑧 → (𝐺𝑧) = (0g𝑌))))
66 csbov12g 7184 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = (𝑧 / 𝑛(𝑛 𝑀)(.r𝐴)𝑧 / 𝑛(𝑈‘(𝐺𝑛))))
67 csbov1g 7185 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑛 𝑀) = (𝑧 / 𝑛𝑛 𝑀))
68 csbvarg 4355 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℕ0𝑧 / 𝑛𝑛 = 𝑧)
6968oveq1d 7155 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0 → (𝑧 / 𝑛𝑛 𝑀) = (𝑧 𝑀))
7067, 69eqtrd 2857 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑛 𝑀) = (𝑧 𝑀))
71 csbfv2g 6696 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑈‘(𝐺𝑛)) = (𝑈𝑧 / 𝑛(𝐺𝑛)))
72 csbfv 6697 . . . . . . . . . . . . . . . . . . . . 21 𝑧 / 𝑛(𝐺𝑛) = (𝐺𝑧)
7372a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝐺𝑛) = (𝐺𝑧))
7473fveq2d 6656 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℕ0 → (𝑈𝑧 / 𝑛(𝐺𝑛)) = (𝑈‘(𝐺𝑧)))
7571, 74eqtrd 2857 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℕ0𝑧 / 𝑛(𝑈‘(𝐺𝑛)) = (𝑈‘(𝐺𝑧)))
7670, 75oveq12d 7158 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℕ0 → (𝑧 / 𝑛(𝑛 𝑀)(.r𝐴)𝑧 / 𝑛(𝑈‘(𝐺𝑛))) = ((𝑧 𝑀)(.r𝐴)(𝑈‘(𝐺𝑧))))
7766, 76eqtrd 2857 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ℕ0𝑧 / 𝑛((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛))) = ((𝑧 𝑀)(.r𝐴)(𝑈‘(𝐺𝑧))))
7877ad2antlr 726 . . . . . . . . . . . . . . 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 6652 . . . . . . . . . . . . . . . . 17 ((𝐺𝑧) = (0g𝑌) → (𝑈‘(𝐺𝑧)) = (𝑈‘(0g𝑌)))
802, 5jca 515 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
8180adantr 484 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
82 eqid 2822 . . . . . . . . . . . . . . . . . . . 20 (0g𝑌) = (0g𝑌)
839, 33, 39, 40, 8, 82m2cpminv0 21364 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑈‘(0g𝑌)) = (0g𝐴))
8481, 83syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) → (𝑈‘(0g𝑌)) = (0g𝐴))
8584ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝑈‘(0g𝑌)) = (0g𝐴))
8679, 85sylan9eqr 2879 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → (𝑈‘(𝐺𝑧)) = (0g𝐴))
8786oveq2d 7156 . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝐴 ∈ Ring)
8923ad3antrrr 729 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (mulGrp‘𝐴) ∈ Mnd)
90 simpr 488 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝑧 ∈ ℕ0)
9126adantr 484 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝑀𝐵)
9228, 29mulgnn0cl 18235 . . . . . . . . . . . . . . . . . . 19 (((mulGrp‘𝐴) ∈ Mnd ∧ 𝑧 ∈ ℕ0𝑀𝐵) → (𝑧 𝑀) ∈ 𝐵)
9389, 90, 91, 92syl3anc 1368 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝑧 𝑀) ∈ 𝐵)
9488, 93jca 515 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝐴 ∈ Ring ∧ (𝑧 𝑀) ∈ 𝐵))
9594adantr 484 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺𝑧) = (0g𝑌)) → (𝐴 ∈ Ring ∧ (𝑧 𝑀) ∈ 𝐵))
967, 50, 8ringrz 19332 . . . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . 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 714 . . . . . . . . . . . 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 3169 . . . . . . . . . 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 3260 . . . . . . . . 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 13360 . . . . . 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 19026 . . . . 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 21356 . . . . 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 1368 . . . 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 21295 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
1112, 5, 110syl2anc 587 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
112 ringmnd 19298 . . . . . . . . 9 (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
113111, 112syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Mnd)
114113ad2antrr 725 . . . . . . 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 21333 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑌))
1173, 6, 116syl2anc 587 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑇 ∈ (𝐴 GrpHom 𝑌))
118 ghmmhm 18359 . . . . . . . 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 729 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
1214, 34sylan2 595 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
1221213adant3 1129 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
123122ad3antrrr 729 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
124123, 48ffvelrnd 6834 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺𝑛)) ∈ 𝐵)
125120, 31, 124, 51syl3anc 1368 . . . . . . 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 19051 . . . . . 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 21337 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑌))
1281273adant3 1129 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑇 ∈ (𝐴 RingHom 𝑌))
129128ad3antrrr 729 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑇 ∈ (𝐴 RingHom 𝑌))
1307, 50, 41rhmmul 19473 . . . . . . . . . 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 1368 . . . . . . . . 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 21336 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
1331323adant3 1129 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
134133ad3antrrr 729 . . . . . . . . . . 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 18259 . . . . . . . . . . 11 ((𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)) ∧ 𝑛 ∈ ℕ0𝑀𝐵) → (𝑇‘(𝑛 𝑀)) = (𝑛𝐸(𝑇𝑀)))
137134, 25, 27, 136syl3anc 1368 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑛 𝑀)) = (𝑛𝐸(𝑇𝑀)))
1382ad3antrrr 729 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
1395ad3antrrr 729 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
14032, 33, 44m2cpminvid2 21358 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐺𝑛) ∈ (𝑁 ConstPolyMat 𝑅)) → (𝑇‘(𝑈‘(𝐺𝑛))) = (𝐺𝑛))
141138, 139, 48, 140syl3anc 1368 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑈‘(𝐺𝑛))) = (𝐺𝑛))
142137, 141oveq12d 7158 . . . . . . . . 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 2857 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘((𝑛 𝑀)(.r𝐴)(𝑈‘(𝐺𝑛)))) = ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))
144143mpteq2dva 5137 . . . . . . 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 7156 . . . . . 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 2859 . . . . 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 6656 . . . 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 2859 . . 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 2879 . 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 21490 . 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 3264 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wral 3130  wrex 3131  Vcvv 3469  csb 3855  ifcif 4439   class class class wbr 5042  cmpt 5122  wf 6330  cfv 6334  (class class class)co 7140  m cmap 8393  Fincfn 8496   finSupp cfsupp 8821  0cc0 10526  1c1 10527   + caddc 10529   < clt 10664  cmin 10859  cn 11625  0cn0 11885  ...cfz 12885  Basecbs 16474  .rcmulr 16557   ·𝑠 cvsca 16560  0gc0g 16704   Σg cgsu 16705  Mndcmnd 17902   MndHom cmhm 17945  -gcsg 18096  .gcmg 18215   GrpHom cghm 18346  CMndccmn 18897  mulGrpcmgp 19230  1rcur 19242  Ringcrg 19288  CRingccrg 19289   RingHom crh 19458  Poly1cpl1 20804  coe1cco1 20805   Mat cmat 21010   ConstPolyMat ccpmat 21306   matToPolyMat cmat2pmat 21307   cPolyMatToMat ccpmat2mat 21308   CharPlyMat cchpmat 21429
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-addf 10605  ax-mulf 10606
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-xor 1503  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-ot 4548  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-of 7394  df-ofr 7395  df-om 7566  df-1st 7675  df-2nd 7676  df-supp 7818  df-tpos 7879  df-cur 7920  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  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 12886  df-fzo 13029  df-seq 13365  df-exp 13426  df-hash 13687  df-word 13858  df-lsw 13906  df-concat 13914  df-s1 13941  df-substr 13994  df-pfx 14024  df-splice 14103  df-reverse 14112  df-s2 14201  df-struct 16476  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-ress 16482  df-plusg 16569  df-mulr 16570  df-starv 16571  df-sca 16572  df-vsca 16573  df-ip 16574  df-tset 16575  df-ple 16576  df-ds 16578  df-unif 16579  df-hom 16580  df-cco 16581  df-0g 16706  df-gsum 16707  df-prds 16712  df-pws 16714  df-mre 16848  df-mrc 16849  df-acs 16851  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-mhm 17947  df-submnd 17948  df-efmnd 18025  df-grp 18097  df-minusg 18098  df-sbg 18099  df-mulg 18216  df-subg 18267  df-ghm 18347  df-gim 18390  df-cntz 18438  df-oppg 18465  df-symg 18487  df-pmtr 18561  df-psgn 18610  df-evpm 18611  df-cmn 18899  df-abl 18900  df-mgp 19231  df-ur 19243  df-srg 19247  df-ring 19290  df-cring 19291  df-oppr 19367  df-dvdsr 19385  df-unit 19386  df-invr 19416  df-dvr 19427  df-rnghom 19461  df-drng 19495  df-subrg 19524  df-lmod 19627  df-lss 19695  df-sra 19935  df-rgmod 19936  df-cnfld 20090  df-zring 20162  df-zrh 20195  df-dsmm 20419  df-frlm 20434  df-assa 20540  df-ascl 20542  df-psr 20592  df-mvr 20593  df-mpl 20594  df-opsr 20596  df-psr1 20807  df-vr1 20808  df-ply1 20809  df-coe1 20810  df-mamu 20989  df-mat 21011  df-mdet 21188  df-madu 21237  df-cpmat 21309  df-mat2pmat 21310  df-cpmat2mat 21311  df-decpmat 21366  df-pm2mp 21396  df-chpmat 21430
This theorem is referenced by:  cayleyhamilton0  21492  cayleyhamiltonALT  21494
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