Theorem cayhamlem4 22832

Description: Lemma for cayleyhamilton 22834. (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 = (Poly1‘𝑅)
chcoeffeq.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
chcoeffeq.r	⊢ × = (.r‘𝑌)
chcoeffeq.s	⊢ − = (-g‘𝑌)
chcoeffeq.0	⊢ 0 = (0g‘𝑌)
chcoeffeq.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
chcoeffeq.c	⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
chcoeffeq.k	⊢ 𝐾 = (𝐶‘𝑀)
chcoeffeq.g	⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
chcoeffeq.w	⊢ 𝑊 = (Base‘𝑌)
chcoeffeq.1	⊢ 1 = (1r‘𝐴)
chcoeffeq.m	⊢ ∗ = ( ·𝑠 ‘𝐴)
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 ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))

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.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))))
2		simp1 1136	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
3	2	ad2antrr 726	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑁 ∈ Fin)
4		crngring 20180	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5	4	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
6	5	ad2antrr 726	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑅 ∈ Ring)
7		chcoeffeq.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
8		eqid 2736	. . . . . 6 ⊢ (0g‘𝐴) = (0g‘𝐴)
9		chcoeffeq.a	. . . . . . . . . . 11 ⊢ 𝐴 = (𝑁 Mat 𝑅)
10	9	matring 22387	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
11	4, 10	sylan2 593	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
12		ringcmn 20217	. . . . . . . . 9 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
13	11, 12	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ CMnd)
14	13	3adant3 1132	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ CMnd)
15	14	ad2antrr 726	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐴 ∈ CMnd)
16		nn0ex 12407	. . . . . . 7 ⊢ ℕ0 ∈ V
17	16	a1i 11	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → ℕ0 ∈ V)
18	3, 6, 10	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐴 ∈ Ring)
19	18	adantr 480	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
20		eqid 2736	. . . . . . . . . 10 ⊢ (mulGrp‘𝐴) = (mulGrp‘𝐴)
21	20, 7	mgpbas 20080	. . . . . . . . 9 ⊢ 𝐵 = (Base‘(mulGrp‘𝐴))
22		cayhamlem.e1	. . . . . . . . 9 ⊢ ↑ = (.g‘(mulGrp‘𝐴))
23	2, 5, 10	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring)
24	20	ringmgp 20174	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Ring → (mulGrp‘𝐴) ∈ Mnd)
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝐴) ∈ Mnd)
26	25	ad3antrrr 730	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (mulGrp‘𝐴) ∈ Mnd)
27		simpr 484	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
28		simpll3 1215	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑀 ∈ 𝐵)
29	28	adantr 480	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ 𝐵)
30	21, 22, 26, 27, 29	mulgnn0cld 19025	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (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)
33	9, 7, 31, 32	cpm2mf 22696	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
34	2, 5, 33	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
35	34	ad3antrrr 730	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
36		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑠 ∈ ℕ)
37		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑏 ∈ (𝐵 ↑m (0...𝑠)))
38		chcoeffeq.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘𝑅)
39		chcoeffeq.y	. . . . . . . . . . . 12 ⊢ 𝑌 = (𝑁 Mat 𝑃)
40		chcoeffeq.r	. . . . . . . . . . . 12 ⊢ × = (.r‘𝑌)
41		chcoeffeq.s	. . . . . . . . . . . 12 ⊢ − = (-g‘𝑌)
42		chcoeffeq.0	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝑌)
43		chcoeffeq.t	. . . . . . . . . . . 12 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
44		chcoeffeq.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
45	9, 7, 38, 39, 40, 41, 42, 43, 44, 31	chfacfisfcpmat 22799	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
46	3, 6, 28, 36, 37, 45	syl32anc 1380	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
47	46	ffvelcdmda 7029	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝐺‘𝑛) ∈ (𝑁 ConstPolyMat 𝑅))
48	35, 47	ffvelcdmd 7030	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺‘𝑛)) ∈ 𝐵)
49		eqid 2736	. . . . . . . . 9 ⊢ (.r‘𝐴) = (.r‘𝐴)
50	7, 49	ringcl 20185	. . . . . . . 8 ⊢ ((𝐴 ∈ Ring ∧ (𝑛 ↑ 𝑀) ∈ 𝐵 ∧ (𝑈‘(𝐺‘𝑛)) ∈ 𝐵) → ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) ∈ 𝐵)
51	19, 30, 48, 50	syl3anc 1373	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) ∈ 𝐵)
52	51	fmpttd 7060	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))):ℕ0⟶𝐵)
53		fvexd 6849	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (0g‘𝐴) ∈ V)
54		ovexd 7393	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) ∈ V)
55	9, 7, 38, 39, 40, 41, 42, 43, 44	chfacffsupp 22800	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺 finSupp (0g‘𝑌))
56	55	anassrs 467	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐺 finSupp (0g‘𝑌))
57		ovex 7391	. . . . . . . . . . . . 13 ⊢ (𝑁 ConstPolyMat 𝑅) ∈ V
58	57, 16	pm3.2i 470	. . . . . . . . . . . 12 ⊢ ((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ0 ∈ V)
59		elmapg 8776	. . . . . . . . . . . 12 ⊢ (((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ0 ∈ V) → (𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m ℕ0) ↔ 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)))
60	58, 59	mp1i 13	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m ℕ0) ↔ 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)))
61	46, 60	mpbird 257	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m ℕ0))
62		fvex 6847	. . . . . . . . . 10 ⊢ (0g‘𝑌) ∈ V
63		fsuppmapnn0ub 13918	. . . . . . . . . 10 ⊢ ((𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m ℕ0) ∧ (0g‘𝑌) ∈ V) → (𝐺 finSupp (0g‘𝑌) → ∃𝑤 ∈ ℕ0 ∀𝑧 ∈ ℕ0 (𝑤 < 𝑧 → (𝐺‘𝑧) = (0g‘𝑌))))
64	61, 62, 63	sylancl 586	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝐺 finSupp (0g‘𝑌) → ∃𝑤 ∈ ℕ0 ∀𝑧 ∈ ℕ0 (𝑤 < 𝑧 → (𝐺‘𝑧) = (0g‘𝑌))))
65		csbov12g 7404	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (⦋𝑧 / 𝑛⦌(𝑛 ↑ 𝑀)(.r‘𝐴)⦋𝑧 / 𝑛⦌(𝑈‘(𝐺‘𝑛))))
66		csbov1g 7405	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑛 ↑ 𝑀) = (⦋𝑧 / 𝑛⦌𝑛 ↑ 𝑀))
67		csbvarg 4386	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌𝑛 = 𝑧)
68	67	oveq1d 7373	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → (⦋𝑧 / 𝑛⦌𝑛 ↑ 𝑀) = (𝑧 ↑ 𝑀))
69	66, 68	eqtrd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑛 ↑ 𝑀) = (𝑧 ↑ 𝑀))
70		csbfv2g 6880	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑈‘(𝐺‘𝑛)) = (𝑈‘⦋𝑧 / 𝑛⦌(𝐺‘𝑛)))
71		csbfv 6881	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⦋𝑧 / 𝑛⦌(𝐺‘𝑛) = (𝐺‘𝑧)
72	71	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝐺‘𝑛) = (𝐺‘𝑧))
73	72	fveq2d 6838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → (𝑈‘⦋𝑧 / 𝑛⦌(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑧)))
74	70, 73	eqtrd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑈‘(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑧)))
75	69, 74	oveq12d 7376	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℕ0 → (⦋𝑧 / 𝑛⦌(𝑛 ↑ 𝑀)(.r‘𝐴)⦋𝑧 / 𝑛⦌(𝑈‘(𝐺‘𝑛))) = ((𝑧 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑧))))
76	65, 75	eqtrd 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = ((𝑧 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑧))))
77	76	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺‘𝑧) = (0g‘𝑌)) → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = ((𝑧 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑧))))
78		fveq2 6834	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑧) = (0g‘𝑌) → (𝑈‘(𝐺‘𝑧)) = (𝑈‘(0g‘𝑌)))
79	2, 5	jca 511	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
80	79	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
81		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0g‘𝑌) = (0g‘𝑌)
82	9, 32, 38, 39, 8, 81	m2cpminv0 22705	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑈‘(0g‘𝑌)) = (0g‘𝐴))
83	80, 82	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) → (𝑈‘(0g‘𝑌)) = (0g‘𝐴))
84	83	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝑈‘(0g‘𝑌)) = (0g‘𝐴))
85	78, 84	sylan9eqr 2793	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺‘𝑧) = (0g‘𝑌)) → (𝑈‘(𝐺‘𝑧)) = (0g‘𝐴))
86	85	oveq2d 7374	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺‘𝑧) = (0g‘𝑌)) → ((𝑧 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑧))) = ((𝑧 ↑ 𝑀)(.r‘𝐴)(0g‘𝐴)))
87	18	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝐴 ∈ Ring)
88	25	ad3antrrr 730	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (mulGrp‘𝐴) ∈ Mnd)
89		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝑧 ∈ ℕ0)
90	28	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝑀 ∈ 𝐵)
91	21, 22, 88, 89, 90	mulgnn0cld 19025	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝑧 ↑ 𝑀) ∈ 𝐵)
92	87, 91	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝐴 ∈ Ring ∧ (𝑧 ↑ 𝑀) ∈ 𝐵))
93	92	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺‘𝑧) = (0g‘𝑌)) → (𝐴 ∈ Ring ∧ (𝑧 ↑ 𝑀) ∈ 𝐵))
94	7, 49, 8	ringrz 20229	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Ring ∧ (𝑧 ↑ 𝑀) ∈ 𝐵) → ((𝑧 ↑ 𝑀)(.r‘𝐴)(0g‘𝐴)) = (0g‘𝐴))
95	93, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺‘𝑧) = (0g‘𝑌)) → ((𝑧 ↑ 𝑀)(.r‘𝐴)(0g‘𝐴)) = (0g‘𝐴))
96	77, 86, 95	3eqtrd 2775	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺‘𝑧) = (0g‘𝑌)) → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴))
97	96	ex 412	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → ((𝐺‘𝑧) = (0g‘𝑌) → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴)))
98	97	adantlr 715	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑤 ∈ ℕ0) ∧ 𝑧 ∈ ℕ0) → ((𝐺‘𝑧) = (0g‘𝑌) → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴)))
99	98	imim2d 57	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑤 ∈ ℕ0) ∧ 𝑧 ∈ ℕ0) → ((𝑤 < 𝑧 → (𝐺‘𝑧) = (0g‘𝑌)) → (𝑤 < 𝑧 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴))))
100	99	ralimdva 3148	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑤 ∈ ℕ0) → (∀𝑧 ∈ ℕ0 (𝑤 < 𝑧 → (𝐺‘𝑧) = (0g‘𝑌)) → ∀𝑧 ∈ ℕ0 (𝑤 < 𝑧 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴))))
101	100	reximdva 3149	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (∃𝑤 ∈ ℕ0 ∀𝑧 ∈ ℕ0 (𝑤 < 𝑧 → (𝐺‘𝑧) = (0g‘𝑌)) → ∃𝑤 ∈ ℕ0 ∀𝑧 ∈ ℕ0 (𝑤 < 𝑧 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴))))
102	64, 101	syld 47	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝐺 finSupp (0g‘𝑌) → ∃𝑤 ∈ ℕ0 ∀𝑧 ∈ ℕ0 (𝑤 < 𝑧 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴))))
103	56, 102	mpd 15	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → ∃𝑤 ∈ ℕ0 ∀𝑧 ∈ ℕ0 (𝑤 < 𝑧 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴)))
104	53, 54, 103	mptnn0fsupp 13920	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))) finSupp (0g‘𝐴))
105	7, 8, 15, 17, 52, 104	gsumcl 19844	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))) ∈ 𝐵)
106	32, 9, 7, 43	m2cpminvid 22697	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))) ∈ 𝐵) → (𝑈‘(𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))))
107	3, 6, 105, 106	syl3anc 1373	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑈‘(𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))))
108	38, 39	pmatring 22636	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
109	2, 5, 108	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
110		ringmnd 20178	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
111	109, 110	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Mnd)
112	111	ad2antrr 726	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑌 ∈ Mnd)
113		chcoeffeq.w	. . . . . . . . . 10 ⊢ 𝑊 = (Base‘𝑌)
114	43, 9, 7, 38, 39, 113	mat2pmatghm 22674	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑌))
115	3, 6, 114	syl2anc 584	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑇 ∈ (𝐴 GrpHom 𝑌))
116		ghmmhm 19155	. . . . . . . 8 ⊢ (𝑇 ∈ (𝐴 GrpHom 𝑌) → 𝑇 ∈ (𝐴 MndHom 𝑌))
117	115, 116	syl 17	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑇 ∈ (𝐴 MndHom 𝑌))
118	23	ad3antrrr 730	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
119	4, 33	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
120	119	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
121	120	ad3antrrr 730	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
122	121, 47	ffvelcdmd 7030	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺‘𝑛)) ∈ 𝐵)
123	118, 30, 122, 50	syl3anc 1373	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) ∈ 𝐵)
124	7, 8, 15, 112, 17, 117, 123, 104	gsummptmhm 19869	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑌 Σg (𝑛 ∈ ℕ0 ↦ (𝑇‘((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) = (𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))))))
125	43, 9, 7, 38, 39, 113	mat2pmatrhm 22678	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑌))
126	125	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑇 ∈ (𝐴 RingHom 𝑌))
127	126	ad3antrrr 730	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑇 ∈ (𝐴 RingHom 𝑌))
128	7, 49, 40	rhmmul 20421	. . . . . . . . . 10 ⊢ ((𝑇 ∈ (𝐴 RingHom 𝑌) ∧ (𝑛 ↑ 𝑀) ∈ 𝐵 ∧ (𝑈‘(𝐺‘𝑛)) ∈ 𝐵) → (𝑇‘((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))) = ((𝑇‘(𝑛 ↑ 𝑀)) × (𝑇‘(𝑈‘(𝐺‘𝑛)))))
129	127, 30, 122, 128	syl3anc 1373	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))) = ((𝑇‘(𝑛 ↑ 𝑀)) × (𝑇‘(𝑈‘(𝐺‘𝑛)))))
130	43, 9, 7, 38, 39, 113	mat2pmatmhm 22677	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
131	130	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
132	131	ad3antrrr 730	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
133		cayhamlem.e2	. . . . . . . . . . . 12 ⊢ 𝐸 = (.g‘(mulGrp‘𝑌))
134	21, 22, 133	mhmmulg 19045	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)) ∧ 𝑛 ∈ ℕ0 ∧ 𝑀 ∈ 𝐵) → (𝑇‘(𝑛 ↑ 𝑀)) = (𝑛𝐸(𝑇‘𝑀)))
135	132, 27, 29, 134	syl3anc 1373	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑛 ↑ 𝑀)) = (𝑛𝐸(𝑇‘𝑀)))
136	2	ad3antrrr 730	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
137	5	ad3antrrr 730	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
138	31, 32, 43	m2cpminvid2 22699	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐺‘𝑛) ∈ (𝑁 ConstPolyMat 𝑅)) → (𝑇‘(𝑈‘(𝐺‘𝑛))) = (𝐺‘𝑛))
139	136, 137, 47, 138	syl3anc 1373	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑈‘(𝐺‘𝑛))) = (𝐺‘𝑛))
140	135, 139	oveq12d 7376	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑇‘(𝑛 ↑ 𝑀)) × (𝑇‘(𝑈‘(𝐺‘𝑛)))) = ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))
141	129, 140	eqtrd 2771	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))) = ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))
142	141	mpteq2dva 5191	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ (𝑇‘((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))) = (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))
143	142	oveq2d 7374	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑌 Σg (𝑛 ∈ ℕ0 ↦ (𝑇‘((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))
144	124, 143	eqtr3d 2773	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))
145	144	fveq2d 6838	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑈‘(𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))
146	107, 145	eqtr3d 2773	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))
147	1, 146	sylan9eqr 2793	. 2 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))
148		chcoeffeq.c	. . 3 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
149		chcoeffeq.k	. . 3 ⊢ 𝐾 = (𝐶‘𝑀)
150		chcoeffeq.1	. . 3 ⊢ 1 = (1r‘𝐴)
151		chcoeffeq.m	. . 3 ⊢ ∗ = ( ·𝑠 ‘𝐴)
152	9, 7, 38, 39, 40, 41, 42, 43, 148, 149, 44, 113, 150, 151, 32, 22, 49	cayhamlem3 22831	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))))
153	147, 152	reximddv2 3195	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  Vcvv 3440  ⦋csb 3849  ifcif 4479   class class class wbr 5098   ↦ cmpt 5179  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763  Fincfn 8883   finSupp cfsupp 9264  0cc0 11026  1c1 11027   + caddc 11029   < clt 11166   − cmin 11364  ℕcn 12145  ℕ₀cn0 12401  ...cfz 13423  Basecbs 17136  ._rcmulr 17178   ·_𝑠 cvsca 17181  0_gc0g 17359   Σ_g cgsu 17360  Mndcmnd 18659   MndHom cmhm 18706  -_gcsg 18865  ._gcmg 18997   GrpHom cghm 19141  CMndccmn 19709  mulGrpcmgp 20075  1_rcur 20116  Ringcrg 20168  CRingccrg 20169   RingHom crh 20405  Poly₁cpl1 22117  coe₁cco1 22118   Mat cmat 22351   ConstPolyMat ccpmat 22647   matToPolyMat cmat2pmat 22648   cPolyMatToMat ccpmat2mat 22649   CharPlyMat cchpmat 22770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-addf 11105  ax-mulf 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1513  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ofr 7623  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-tpos 8168  df-cur 8209  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-sup 9345  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-xnn0 12475  df-z 12489  df-dec 12608  df-uz 12752  df-rp 12906  df-fz 13424  df-fzo 13571  df-seq 13925  df-exp 13985  df-hash 14254  df-word 14437  df-lsw 14486  df-concat 14494  df-s1 14520  df-substr 14565  df-pfx 14595  df-splice 14673  df-reverse 14682  df-s2 14771  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-starv 17192  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-unif 17200  df-hom 17201  df-cco 17202  df-0g 17361  df-gsum 17362  df-prds 17367  df-pws 17369  df-mre 17505  df-mrc 17506  df-acs 17508  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-submnd 18709  df-efmnd 18794  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-subg 19053  df-ghm 19142  df-gim 19188  df-cntz 19246  df-oppg 19275  df-symg 19299  df-pmtr 19371  df-psgn 19420  df-evpm 19421  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-srg 20122  df-ring 20170  df-cring 20171  df-oppr 20273  df-dvdsr 20293  df-unit 20294  df-invr 20324  df-dvr 20337  df-rhm 20408  df-subrng 20479  df-subrg 20503  df-drng 20664  df-lmod 20813  df-lss 20883  df-sra 21125  df-rgmod 21126  df-cnfld 21310  df-zring 21402  df-zrh 21458  df-dsmm 21687  df-frlm 21702  df-assa 21808  df-ascl 21810  df-psr 21865  df-mvr 21866  df-mpl 21867  df-opsr 21869  df-psr1 22120  df-vr1 22121  df-ply1 22122  df-coe1 22123  df-mamu 22335  df-mat 22352  df-mdet 22529  df-madu 22578  df-cpmat 22650  df-mat2pmat 22651  df-cpmat2mat 22652  df-decpmat 22707  df-pm2mp 22737  df-chpmat 22771

This theorem is referenced by:  cayleyhamilton0  22833  cayleyhamiltonALT  22835