Theorem cayhamlem4 22915

Description: Lemma for cayleyhamilton 22917. (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 725	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑁 ∈ Fin)
4		crngring 20272	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5	4	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
6	5	ad2antrr 725	. . . . 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 2740	. . . . . 6 ⊢ (0g‘𝐴) = (0g‘𝐴)
9		chcoeffeq.a	. . . . . . . . . . 11 ⊢ 𝐴 = (𝑁 Mat 𝑅)
10	9	matring 22470	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
11	4, 10	sylan2 592	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
12		ringcmn 20305	. . . . . . . . 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 725	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐴 ∈ CMnd)
16		nn0ex 12559	. . . . . . 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 583	. . . . . . . . 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 2740	. . . . . . . . . 10 ⊢ (mulGrp‘𝐴) = (mulGrp‘𝐴)
21	20, 7	mgpbas 20167	. . . . . . . . 9 ⊢ 𝐵 = (Base‘(mulGrp‘𝐴))
22		cayhamlem.e1	. . . . . . . . 9 ⊢ ↑ = (.g‘(mulGrp‘𝐴))
23	2, 5, 10	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring)
24	20	ringmgp 20266	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Ring → (mulGrp‘𝐴) ∈ Mnd)
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝐴) ∈ Mnd)
26	25	ad3antrrr 729	. . . . . . . . 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 1214	. . . . . . . . . 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 19135	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑀) ∈ 𝐵)
31		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
32		chcoeffeq.u	. . . . . . . . . . . 12 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
33	9, 7, 31, 32	cpm2mf 22779	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
34	2, 5, 33	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
35	34	ad3antrrr 729	. . . . . . . . 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 22882	. . . . . . . . . . 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 1378	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
47	46	ffvelcdmda 7118	. . . . . . . . 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 7119	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺‘𝑛)) ∈ 𝐵)
49		eqid 2740	. . . . . . . . 9 ⊢ (.r‘𝐴) = (.r‘𝐴)
50	7, 49	ringcl 20277	. . . . . . . 8 ⊢ ((𝐴 ∈ Ring ∧ (𝑛 ↑ 𝑀) ∈ 𝐵 ∧ (𝑈‘(𝐺‘𝑛)) ∈ 𝐵) → ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) ∈ 𝐵)
51	19, 30, 48, 50	syl3anc 1371	. . . . . . 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 7149	. . . . . 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 6935	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (0g‘𝐴) ∈ V)
54		ovexd 7483	. . . . . . 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 22883	. . . . . . . . 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 7481	. . . . . . . . . . . . 13 ⊢ (𝑁 ConstPolyMat 𝑅) ∈ V
58	57, 16	pm3.2i 470	. . . . . . . . . . . 12 ⊢ ((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ0 ∈ V)
59		elmapg 8897	. . . . . . . . . . . 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 6933	. . . . . . . . . 10 ⊢ (0g‘𝑌) ∈ V
63		fsuppmapnn0ub 14046	. . . . . . . . . 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 585	. . . . . . . . 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 7494	. . . . . . . . . . . . . . . . 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 7495	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑛 ↑ 𝑀) = (⦋𝑧 / 𝑛⦌𝑛 ↑ 𝑀))
67		csbvarg 4457	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌𝑛 = 𝑧)
68	67	oveq1d 7463	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → (⦋𝑧 / 𝑛⦌𝑛 ↑ 𝑀) = (𝑧 ↑ 𝑀))
69	66, 68	eqtrd 2780	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑛 ↑ 𝑀) = (𝑧 ↑ 𝑀))
70		csbfv2g 6969	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑈‘(𝐺‘𝑛)) = (𝑈‘⦋𝑧 / 𝑛⦌(𝐺‘𝑛)))
71		csbfv 6970	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⦋𝑧 / 𝑛⦌(𝐺‘𝑛) = (𝐺‘𝑧)
72	71	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝐺‘𝑛) = (𝐺‘𝑧))
73	72	fveq2d 6924	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ0 → (𝑈‘⦋𝑧 / 𝑛⦌(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑧)))
74	70, 73	eqtrd 2780	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌(𝑈‘(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑧)))
75	69, 74	oveq12d 7466	. . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ℕ0 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = ((𝑧 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑧))))
77	76	ad2antlr 726	. . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0g‘𝑌) = (0g‘𝑌)
82	9, 32, 38, 39, 8, 81	m2cpminv0 22788	. . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . 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 2802	. . . . . . . . . . . . . . . 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 7464	. . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . . . 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 19135	. . . . . . . . . . . . . . . . . 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 20317	. . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . 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 714	. . . . . . . . . . . 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 3173	. . . . . . . . . 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 3174	. . . . . . . . 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 14048	. . . . . 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 19957	. . . . 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 22780	. . . . 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 1371	. . . 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 22719	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
109	2, 5, 108	syl2anc 583	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
110		ringmnd 20270	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
111	109, 110	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Mnd)
112	111	ad2antrr 725	. . . . . . 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 22757	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑌))
115	3, 6, 114	syl2anc 583	. . . . . . . 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 729	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
119	4, 33	sylan2 592	. . . . . . . . . . 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 729	. . . . . . . . 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 7119	. . . . . . . 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 1371	. . . . . . 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 19982	. . . . . 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 22761	. . . . . . . . . . . 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 729	. . . . . . . . . 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 20512	. . . . . . . . . 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 1371	. . . . . . . . 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 22760	. . . . . . . . . . . . 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 729	. . . . . . . . . . 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 19155	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)) ∧ 𝑛 ∈ ℕ0 ∧ 𝑀 ∈ 𝐵) → (𝑇‘(𝑛 ↑ 𝑀)) = (𝑛𝐸(𝑇‘𝑀)))
135	132, 27, 29, 134	syl3anc 1371	. . . . . . . . . 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 729	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
137	5	ad3antrrr 729	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
138	31, 32, 43	m2cpminvid2 22782	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐺‘𝑛) ∈ (𝑁 ConstPolyMat 𝑅)) → (𝑇‘(𝑈‘(𝐺‘𝑛))) = (𝐺‘𝑛))
139	136, 137, 47, 138	syl3anc 1371	. . . . . . . . . 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 7466	. . . . . . . . 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 2780	. . . . . . . 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 5266	. . . . . . 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 7464	. . . . . 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 2782	. . . . 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 6924	. . . 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 2782	. . 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 2802	. 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 22914	. 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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  Vcvv 3488  ⦋csb 3921  ifcif 4548   class class class wbr 5166   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  Fincfn 9003   finSupp cfsupp 9431  0cc0 11184  1c1 11185   + caddc 11187   < clt 11324   − cmin 11520  ℕcn 12293  ℕ₀cn0 12553  ...cfz 13567  Basecbs 17258  ._rcmulr 17312   ·_𝑠 cvsca 17315  0_gc0g 17499   Σ_g cgsu 17500  Mndcmnd 18772   MndHom cmhm 18816  -_gcsg 18975  ._gcmg 19107   GrpHom cghm 19252  CMndccmn 19822  mulGrpcmgp 20161  1_rcur 20208  Ringcrg 20260  CRingccrg 20261   RingHom crh 20495  Poly₁cpl1 22199  coe₁cco1 22200   Mat cmat 22432   ConstPolyMat ccpmat 22730   matToPolyMat cmat2pmat 22731   cPolyMatToMat ccpmat2mat 22732   CharPlyMat cchpmat 22853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-addf 11263  ax-mulf 11264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-xor 1509  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-ofr 7715  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-tpos 8267  df-cur 8308  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-sup 9511  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-xnn0 12626  df-z 12640  df-dec 12759  df-uz 12904  df-rp 13058  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-word 14563  df-lsw 14611  df-concat 14619  df-s1 14644  df-substr 14689  df-pfx 14719  df-splice 14798  df-reverse 14807  df-s2 14897  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-starv 17326  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-unif 17334  df-hom 17335  df-cco 17336  df-0g 17501  df-gsum 17502  df-prds 17507  df-pws 17509  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-efmnd 18904  df-grp 18976  df-minusg 18977  df-sbg 18978  df-mulg 19108  df-subg 19163  df-ghm 19253  df-gim 19299  df-cntz 19357  df-oppg 19386  df-symg 19411  df-pmtr 19484  df-psgn 19533  df-evpm 19534  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-srg 20214  df-ring 20262  df-cring 20263  df-oppr 20360  df-dvdsr 20383  df-unit 20384  df-invr 20414  df-dvr 20427  df-rhm 20498  df-subrng 20572  df-subrg 20597  df-drng 20753  df-lmod 20882  df-lss 20953  df-sra 21195  df-rgmod 21196  df-cnfld 21388  df-zring 21481  df-zrh 21537  df-dsmm 21775  df-frlm 21790  df-assa 21896  df-ascl 21898  df-psr 21952  df-mvr 21953  df-mpl 21954  df-opsr 21956  df-psr1 22202  df-vr1 22203  df-ply1 22204  df-coe1 22205  df-mamu 22416  df-mat 22433  df-mdet 22612  df-madu 22661  df-cpmat 22733  df-mat2pmat 22734  df-cpmat2mat 22735  df-decpmat 22790  df-pm2mp 22820  df-chpmat 22854

This theorem is referenced by:  cayleyhamilton0  22916  cayleyhamiltonALT  22918