Theorem cayhamlem4 22782

Description: Lemma for cayleyhamilton 22784. (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 20161	. . . . . . 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 2730	. . . . . 6 ⊢ (0g‘𝐴) = (0g‘𝐴)
9		chcoeffeq.a	. . . . . . . . . . 11 ⊢ 𝐴 = (𝑁 Mat 𝑅)
10	9	matring 22337	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
11	4, 10	sylan2 593	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
12		ringcmn 20198	. . . . . . . . 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 12455	. . . . . . 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 2730	. . . . . . . . . 10 ⊢ (mulGrp‘𝐴) = (mulGrp‘𝐴)
21	20, 7	mgpbas 20061	. . . . . . . . 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 20155	. . . . . . . . . . 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 19034	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑀) ∈ 𝐵)
31		eqid 2730	. . . . . . . . . . . 12 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
32		chcoeffeq.u	. . . . . . . . . . . 12 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
33	9, 7, 31, 32	cpm2mf 22646	. . . . . . . . . . 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 22749	. . . . . . . . . . 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 7059	. . . . . . . . 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 7060	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺‘𝑛)) ∈ 𝐵)
49		eqid 2730	. . . . . . . . 9 ⊢ (.r‘𝐴) = (.r‘𝐴)
50	7, 49	ringcl 20166	. . . . . . . 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 7090	. . . . . 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 6876	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (0g‘𝐴) ∈ V)
54		ovexd 7425	. . . . . . 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 22750	. . . . . . . . 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 7423	. . . . . . . . . . . . 13 ⊢ (𝑁 ConstPolyMat 𝑅) ∈ V
58	57, 16	pm3.2i 470	. . . . . . . . . . . 12 ⊢ ((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ0 ∈ V)
59		elmapg 8815	. . . . . . . . . . . 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 6874	. . . . . . . . . 10 ⊢ (0g‘𝑌) ∈ V
63		fsuppmapnn0ub 13967	. . . . . . . . . 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 7436	. . . . . . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑛 ↑ 𝑀) = (⦋𝑧 / 𝑛⦌𝑛 ↑ 𝑀))
67		csbvarg 4400	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌𝑛 = 𝑧)
68	67	oveq1d 7405	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → (⦋𝑧 / 𝑛⦌𝑛 ↑ 𝑀) = (𝑧 ↑ 𝑀))
69	66, 68	eqtrd 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑛 ↑ 𝑀) = (𝑧 ↑ 𝑀))
70		csbfv2g 6910	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑈‘(𝐺‘𝑛)) = (𝑈‘⦋𝑧 / 𝑛⦌(𝐺‘𝑛)))
71		csbfv 6911	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⦋𝑧 / 𝑛⦌(𝐺‘𝑛) = (𝐺‘𝑧)
72	71	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝐺‘𝑛) = (𝐺‘𝑧))
73	72	fveq2d 6865	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → (𝑈‘⦋𝑧 / 𝑛⦌(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑧)))
74	70, 73	eqtrd 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑈‘(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑧)))
75	69, 74	oveq12d 7408	. . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0g‘𝑌) = (0g‘𝑌)
82	9, 32, 38, 39, 8, 81	m2cpminv0 22655	. . . . . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . . . 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 7406	. . . . . . . . . . . . . . 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 19034	. . . . . . . . . . . . . . . . . 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 20210	. . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . 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 3146	. . . . . . . . . 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 3147	. . . . . . . . 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 13969	. . . . . 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 19852	. . . . 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 22647	. . . . 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 22586	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
109	2, 5, 108	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
110		ringmnd 20159	. . . . . . . . 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 22624	. . . . . . . . 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 19165	. . . . . . . 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 7060	. . . . . . . 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 19877	. . . . . 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 22628	. . . . . . . . . . . 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 20402	. . . . . . . . . 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 22627	. . . . . . . . . . . . 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 19054	. . . . . . . . . . 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 22649	. . . . . . . . . . 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 7408	. . . . . . . . 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 2765	. . . . . . . 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 5203	. . . . . . 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 7406	. . . . . 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 2767	. . . . 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 6865	. . . 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 2767	. . 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 2787	. 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 22781	. 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 3197	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 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450  ⦋csb 3865  ifcif 4491   class class class wbr 5110   ↦ cmpt 5191  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802  Fincfn 8921   finSupp cfsupp 9319  0cc0 11075  1c1 11076   + caddc 11078   < clt 11215   − cmin 11412  ℕcn 12193  ℕ₀cn0 12449  ...cfz 13475  Basecbs 17186  ._rcmulr 17228   ·_𝑠 cvsca 17231  0_gc0g 17409   Σ_g cgsu 17410  Mndcmnd 18668   MndHom cmhm 18715  -_gcsg 18874  ._gcmg 19006   GrpHom cghm 19151  CMndccmn 19717  mulGrpcmgp 20056  1_rcur 20097  Ringcrg 20149  CRingccrg 20150   RingHom crh 20385  Poly₁cpl1 22068  coe₁cco1 22069   Mat cmat 22301   ConstPolyMat ccpmat 22597   matToPolyMat cmat2pmat 22598   cPolyMatToMat ccpmat2mat 22599   CharPlyMat cchpmat 22720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-addf 11154  ax-mulf 11155

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1512  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-ofr 7657  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-tpos 8208  df-cur 8249  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-pm 8805  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-sup 9400  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-xnn0 12523  df-z 12537  df-dec 12657  df-uz 12801  df-rp 12959  df-fz 13476  df-fzo 13623  df-seq 13974  df-exp 14034  df-hash 14303  df-word 14486  df-lsw 14535  df-concat 14543  df-s1 14568  df-substr 14613  df-pfx 14643  df-splice 14722  df-reverse 14731  df-s2 14821  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-starv 17242  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-hom 17251  df-cco 17252  df-0g 17411  df-gsum 17412  df-prds 17417  df-pws 17419  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-submnd 18718  df-efmnd 18803  df-grp 18875  df-minusg 18876  df-sbg 18877  df-mulg 19007  df-subg 19062  df-ghm 19152  df-gim 19198  df-cntz 19256  df-oppg 19285  df-symg 19307  df-pmtr 19379  df-psgn 19428  df-evpm 19429  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-srg 20103  df-ring 20151  df-cring 20152  df-oppr 20253  df-dvdsr 20273  df-unit 20274  df-invr 20304  df-dvr 20317  df-rhm 20388  df-subrng 20462  df-subrg 20486  df-drng 20647  df-lmod 20775  df-lss 20845  df-sra 21087  df-rgmod 21088  df-cnfld 21272  df-zring 21364  df-zrh 21420  df-dsmm 21648  df-frlm 21663  df-assa 21769  df-ascl 21771  df-psr 21825  df-mvr 21826  df-mpl 21827  df-opsr 21829  df-psr1 22071  df-vr1 22072  df-ply1 22073  df-coe1 22074  df-mamu 22285  df-mat 22302  df-mdet 22479  df-madu 22528  df-cpmat 22600  df-mat2pmat 22601  df-cpmat2mat 22602  df-decpmat 22657  df-pm2mp 22687  df-chpmat 22721

This theorem is referenced by:  cayleyhamilton0  22783  cayleyhamiltonALT  22785