Theorem cayhamlem4 22945

Description: Lemma for cayleyhamilton 22947. (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 1149	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
3	2	ad2antrr 736	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑁 ∈ Fin)
4		crngring 20291	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5	4	3ad2ant2 1147	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
6	5	ad2antrr 736	. . . . 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 2762	. . . . . 6 ⊢ (0g‘𝐴) = (0g‘𝐴)
9		chcoeffeq.a	. . . . . . . . . . 11 ⊢ 𝐴 = (𝑁 Mat 𝑅)
10	9	matring 22500	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
11	4, 10	sylan2 602	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
12		ringcmn 20328	. . . . . . . . 9 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
13	11, 12	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ CMnd)
14	13	3adant3 1145	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ CMnd)
15	14	ad2antrr 736	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐴 ∈ CMnd)
16		nn0ex 12487	. . . . . . 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 593	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐴 ∈ Ring)
19	18	adantr 484	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
20		eqid 2762	. . . . . . . . . 10 ⊢ (mulGrp‘𝐴) = (mulGrp‘𝐴)
21	20, 7	mgpbas 20191	. . . . . . . . 9 ⊢ 𝐵 = (Base‘(mulGrp‘𝐴))
22		cayhamlem.e1	. . . . . . . . 9 ⊢ ↑ = (.g‘(mulGrp‘𝐴))
23	2, 5, 10	syl2anc 593	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring)
24	20	ringmgp 20285	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Ring → (mulGrp‘𝐴) ∈ Mnd)
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝐴) ∈ Mnd)
26	25	ad3antrrr 740	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (mulGrp‘𝐴) ∈ Mnd)
27		simpr 488	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
28		simpll3 1228	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑀 ∈ 𝐵)
29	28	adantr 484	. . . . . . . . 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 19137	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑀) ∈ 𝐵)
31		eqid 2762	. . . . . . . . . . . 12 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
32		chcoeffeq.u	. . . . . . . . . . . 12 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
33	9, 7, 31, 32	cpm2mf 22809	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
34	2, 5, 33	syl2anc 593	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
35	34	ad3antrrr 740	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
36		simplr 778	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑠 ∈ ℕ)
37		simpr 488	. . . . . . . . . . 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 22912	. . . . . . . . . . 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 1397	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
47	46	ffvelcdmda 7065	. . . . . . . . 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 7066	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺‘𝑛)) ∈ 𝐵)
49		eqid 2762	. . . . . . . . 9 ⊢ (.r‘𝐴) = (.r‘𝐴)
50	7, 49	ringcl 20296	. . . . . . . 8 ⊢ ((𝐴 ∈ Ring ∧ (𝑛 ↑ 𝑀) ∈ 𝐵 ∧ (𝑈‘(𝐺‘𝑛)) ∈ 𝐵) → ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) ∈ 𝐵)
51	19, 30, 48, 50	syl3anc 1390	. . . . . . 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 7096	. . . . . 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 6882	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (0g‘𝐴) ∈ V)
54		ovexd 7431	. . . . . . 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 22913	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺 finSupp (0g‘𝑌))
56	55	anassrs 471	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐺 finSupp (0g‘𝑌))
57		ovex 7429	. . . . . . . . . . . . 13 ⊢ (𝑁 ConstPolyMat 𝑅) ∈ V
58	57, 16	pm3.2i 474	. . . . . . . . . . . 12 ⊢ ((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ0 ∈ V)
59		elmapg 8820	. . . . . . . . . . . 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 259	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑m ℕ0))
62		fvex 6880	. . . . . . . . . 10 ⊢ (0g‘𝑌) ∈ V
63		fsuppmapnn0ub 14008	. . . . . . . . . 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 595	. . . . . . . . 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 7442	. . . . . . . . . . . . . . . . 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 7443	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑛 ↑ 𝑀) = (⦋𝑧 / 𝑛⦌𝑛 ↑ 𝑀))
67		csbvarg 4388	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌𝑛 = 𝑧)
68	67	oveq1d 7411	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → (⦋𝑧 / 𝑛⦌𝑛 ↑ 𝑀) = (𝑧 ↑ 𝑀))
69	66, 68	eqtrd 2797	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑛 ↑ 𝑀) = (𝑧 ↑ 𝑀))
70		csbfv2g 6913	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑈‘(𝐺‘𝑛)) = (𝑈‘⦋𝑧 / 𝑛⦌(𝐺‘𝑛)))
71		csbfv 6914	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⦋𝑧 / 𝑛⦌(𝐺‘𝑛) = (𝐺‘𝑧)
72	71	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝐺‘𝑛) = (𝐺‘𝑧))
73	72	fveq2d 6871	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → (𝑈‘⦋𝑧 / 𝑛⦌(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑧)))
74	70, 73	eqtrd 2797	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑈‘(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑧)))
75	69, 74	oveq12d 7414	. . . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = ((𝑧 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑧))))
77	76	ad2antlr 737	. . . . . . . . . . . . . . 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 6867	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑧) = (0g‘𝑌) → (𝑈‘(𝐺‘𝑧)) = (𝑈‘(0g‘𝑌)))
79	2, 5	jca 519	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
80	79	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
81		eqid 2762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0g‘𝑌) = (0g‘𝑌)
82	9, 32, 38, 39, 8, 81	m2cpminv0 22818	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . 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 2819	. . . . . . . . . . . . . . . 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 7412	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝐴 ∈ Ring)
88	25	ad3antrrr 740	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (mulGrp‘𝐴) ∈ Mnd)
89		simpr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝑧 ∈ ℕ0)
90	28	adantr 484	. . . . . . . . . . . . . . . . . . 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 19137	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝑧 ↑ 𝑀) ∈ 𝐵)
92	87, 91	jca 519	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝐴 ∈ Ring ∧ (𝑧 ↑ 𝑀) ∈ 𝐵))
93	92	adantr 484	. . . . . . . . . . . . . . . 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 20340	. . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . 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 416	. . . . . . . . . . . . 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 725	. . . . . . . . . . . 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 3174	. . . . . . . . . 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 3175	. . . . . . . . 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 14010	. . . . . 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 19955	. . . . 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 22810	. . . . 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 1390	. . . 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 22749	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
109	2, 5, 108	syl2anc 593	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
110		ringmnd 20289	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
111	109, 110	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Mnd)
112	111	ad2antrr 736	. . . . . . 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 22787	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑌))
115	3, 6, 114	syl2anc 593	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑇 ∈ (𝐴 GrpHom 𝑌))
116		ghmmhm 19266	. . . . . . . 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 740	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
119	4, 33	sylan2 602	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
120	119	3adant3 1145	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
121	120	ad3antrrr 740	. . . . . . . . 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 7066	. . . . . . . 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 1390	. . . . . . 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 19980	. . . . . 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 22791	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑌))
126	125	3adant3 1145	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑇 ∈ (𝐴 RingHom 𝑌))
127	126	ad3antrrr 740	. . . . . . . . . 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 20531	. . . . . . . . . 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 1390	. . . . . . . . 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 22790	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
131	130	3adant3 1145	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)))
132	131	ad3antrrr 740	. . . . . . . . . . 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 19157	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)) ∧ 𝑛 ∈ ℕ0 ∧ 𝑀 ∈ 𝐵) → (𝑇‘(𝑛 ↑ 𝑀)) = (𝑛𝐸(𝑇‘𝑀)))
135	132, 27, 29, 134	syl3anc 1390	. . . . . . . . . 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 740	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
137	5	ad3antrrr 740	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
138	31, 32, 43	m2cpminvid2 22812	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐺‘𝑛) ∈ (𝑁 ConstPolyMat 𝑅)) → (𝑇‘(𝑈‘(𝐺‘𝑛))) = (𝐺‘𝑛))
139	136, 137, 47, 138	syl3anc 1390	. . . . . . . . . 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 7414	. . . . . . . . 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 2797	. . . . . . . 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 5193	. . . . . . 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 7412	. . . . . 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 2799	. . . . 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 6871	. . . 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 2799	. . 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 2819	. 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 22944	. 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 3221	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 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  Vcvv 3454  ⦋csb 3852  ifcif 4480   class class class wbr 5100   ↦ cmpt 5181  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396   ↑_m cmap 8808  Fincfn 8927   finSupp cfsupp 9307  0cc0 11073  1c1 11074   + caddc 11076   < clt 11216   − cmin 11414  ℕcn 12210  ℕ₀cn0 12481  ...cfz 13512  Basecbs 17245  ._rcmulr 17287   ·_𝑠 cvsca 17290  0_gc0g 17468   Σ_g cgsu 17469  Mndcmnd 18768   MndHom cmhm 18815  -_gcsg 18977  ._gcmg 19109   GrpHom cghm 19253  CMndccmn 19820  mulGrpcmgp 20186  1_rcur 20227  Ringcrg 20279  CRingccrg 20280   RingHom crh 20514  Poly₁cpl1 22236  coe₁cco1 22237   Mat cmat 22464   ConstPolyMat ccpmat 22760   matToPolyMat cmat2pmat 22761   cPolyMatToMat ccpmat2mat 22762   CharPlyMat cchpmat 22883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-addf 11152  ax-mulf 11153

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-xor 1532  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-ofr 7661  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-tpos 8206  df-cur 8247  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-pm 8811  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9308  df-sup 9388  df-oi 9458  df-card 9897  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12482  df-xnn0 12555  df-z 12569  df-dec 12689  df-uz 12840  df-rp 12994  df-fz 13513  df-fzo 13660  df-seq 14015  df-exp 14075  df-hash 14344  df-word 14527  df-lsw 14576  df-concat 14584  df-s1 14610  df-substr 14655  df-pfx 14685  df-splice 14763  df-reverse 14772  df-s2 14861  df-struct 17183  df-sets 17200  df-slot 17218  df-ndx 17230  df-base 17246  df-ress 17267  df-plusg 17299  df-mulr 17300  df-starv 17301  df-sca 17302  df-vsca 17303  df-ip 17304  df-tset 17305  df-ple 17306  df-ds 17308  df-unif 17309  df-hom 17310  df-cco 17311  df-0g 17470  df-gsum 17471  df-prds 17476  df-pws 17478  df-mre 17614  df-mrc 17615  df-acs 17617  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-mhm 18817  df-submnd 18818  df-efmnd 18903  df-grp 18978  df-minusg 18979  df-sbg 18980  df-mulg 19110  df-subg 19165  df-ghm 19254  df-gim 19299  df-cntz 19357  df-oppg 19386  df-symg 19410  df-pmtr 19482  df-psgn 19531  df-evpm 19532  df-cmn 19822  df-abl 19823  df-mgp 20187  df-rng 20199  df-ur 20228  df-srg 20233  df-ring 20281  df-cring 20282  df-oppr 20382  df-dvdsr 20402  df-unit 20403  df-invr 20433  df-dvr 20446  df-rhm 20517  df-subrng 20592  df-subrg 20616  df-drng 20777  df-lmod 20926  df-lss 20996  df-sra 21237  df-rgmod 21238  df-cnfld 21422  df-zring 21496  df-zrh 21552  df-dsmm 21781  df-frlm 21796  df-assa 21902  df-ascl 21904  df-psr 21958  df-mvr 21959  df-mpl 21960  df-opsr 21962  df-psr1 22239  df-vr1 22240  df-ply1 22241  df-coe1 22242  df-mamu 22448  df-mat 22465  df-mdet 22642  df-madu 22691  df-cpmat 22763  df-mat2pmat 22764  df-cpmat2mat 22765  df-decpmat 22820  df-pm2mp 22850  df-chpmat 22884

This theorem is referenced by:  cayleyhamilton0  22946  cayleyhamiltonALT  22948