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Theorem cayhamlem4 22397

Description: Lemma for cayleyhamilton 22399. (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 = (0_g‘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 = (1_r‘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 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σ_g (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 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))))))
2		simp1 1136	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
3	2	ad2antrr 724	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑁 ∈ Fin)
4		crngring 20070	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5	4	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
6	5	ad2antrr 724	. . . . 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 2732	. . . . . 6 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
9		chcoeffeq.a	. . . . . . . . . . 11 ⊢ 𝐴 = (𝑁 Mat 𝑅)
10	9	matring 21952	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
11	4, 10	sylan2 593	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
12		ringcmn 20101	. . . . . . . . 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 724	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝐴 ∈ CMnd)
16		nn0ex 12480	. . . . . . 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 481	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑛 ∈ ℕ₀) → 𝐴 ∈ Ring)
20		eqid 2732	. . . . . . . . . 10 ⊢ (mulGrp‘𝐴) = (mulGrp‘𝐴)
21	20, 7	mgpbas 19995	. . . . . . . . 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 20064	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Ring → (mulGrp‘𝐴) ∈ Mnd)
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝐴) ∈ Mnd)
26	25	ad3antrrr 728	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑛 ∈ ℕ₀) → (mulGrp‘𝐴) ∈ Mnd)
27		simpr 485	. . . . . . . . 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 481	. . . . . . . . 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 18977	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑛 ∈ ℕ₀) → (𝑛 ↑ 𝑀) ∈ 𝐵)
31		eqid 2732	. . . . . . . . . . . 12 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
32		chcoeffeq.u	. . . . . . . . . . . 12 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
33	9, 7, 31, 32	cpm2mf 22261	. . . . . . . . . . 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 728	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑛 ∈ ℕ₀) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
36		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑠 ∈ ℕ)
37		simpr 485	. . . . . . . . . . 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 = (0_g‘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 22364	. . . . . . . . . . 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 7086	. . . . . . . . 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 7087	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑛 ∈ ℕ₀) → (𝑈‘(𝐺‘𝑛)) ∈ 𝐵)
49		eqid 2732	. . . . . . . . 9 ⊢ (._r‘𝐴) = (._r‘𝐴)
50	7, 49	ringcl 20075	. . . . . . . 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 7116	. . . . . 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 6906	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (0_g‘𝐴) ∈ V)
54		ovexd 7446	. . . . . . 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 22365	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺 finSupp (0_g‘𝑌))
56	55	anassrs 468	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝐺 finSupp (0_g‘𝑌))
57		ovex 7444	. . . . . . . . . . . . 13 ⊢ (𝑁 ConstPolyMat 𝑅) ∈ V
58	57, 16	pm3.2i 471	. . . . . . . . . . . 12 ⊢ ((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ₀ ∈ V)
59		elmapg 8835	. . . . . . . . . . . 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 256	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑_m ℕ₀))
62		fvex 6904	. . . . . . . . . 10 ⊢ (0_g‘𝑌) ∈ V
63		fsuppmapnn0ub 13962	. . . . . . . . . 10 ⊢ ((𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑_m ℕ₀) ∧ (0_g‘𝑌) ∈ V) → (𝐺 finSupp (0_g‘𝑌) → ∃𝑤 ∈ ℕ₀ ∀𝑧 ∈ ℕ₀ (𝑤 < 𝑧 → (𝐺‘𝑧) = (0_g‘𝑌))))
64	61, 62, 63	sylancl 586	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝐺 finSupp (0_g‘𝑌) → ∃𝑤 ∈ ℕ₀ ∀𝑧 ∈ ℕ₀ (𝑤 < 𝑧 → (𝐺‘𝑧) = (0_g‘𝑌))))
65		csbov12g 7455	. . . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ₀ → ⦋𝑧 / 𝑛⦌(𝑛 ↑ 𝑀) = (⦋𝑧 / 𝑛⦌𝑛 ↑ 𝑀))
67		csbvarg 4431	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ₀ → ⦋𝑧 / 𝑛⦌𝑛 = 𝑧)
68	67	oveq1d 7426	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ₀ → (⦋𝑧 / 𝑛⦌𝑛 ↑ 𝑀) = (𝑧 ↑ 𝑀))
69	66, 68	eqtrd 2772	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℕ₀ → ⦋𝑧 / 𝑛⦌(𝑛 ↑ 𝑀) = (𝑧 ↑ 𝑀))
70		csbfv2g 6940	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ₀ → ⦋𝑧 / 𝑛⦌(𝑈‘(𝐺‘𝑛)) = (𝑈‘⦋𝑧 / 𝑛⦌(𝐺‘𝑛)))
71		csbfv 6941	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⦋𝑧 / 𝑛⦌(𝐺‘𝑛) = (𝐺‘𝑧)
72	71	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ₀ → ⦋𝑧 / 𝑛⦌(𝐺‘𝑛) = (𝐺‘𝑧))
73	72	fveq2d 6895	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℕ₀ → (𝑈‘⦋𝑧 / 𝑛⦌(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑧)))
74	70, 73	eqtrd 2772	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℕ₀ → ⦋𝑧 / 𝑛⦌(𝑈‘(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑧)))
75	69, 74	oveq12d 7429	. . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ℕ₀ → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))) = ((𝑧 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑧))))
77	76	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑧 ∈ ℕ₀) ∧ (𝐺‘𝑧) = (0_g‘𝑌)) → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))) = ((𝑧 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑧))))
78		fveq2 6891	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑧) = (0_g‘𝑌) → (𝑈‘(𝐺‘𝑧)) = (𝑈‘(0_g‘𝑌)))
79	2, 5	jca 512	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
80	79	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
81		eqid 2732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
82	9, 32, 38, 39, 8, 81	m2cpminv0 22270	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
83	80, 82	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
84	83	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑧 ∈ ℕ₀) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
85	78, 84	sylan9eqr 2794	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑧 ∈ ℕ₀) ∧ (𝐺‘𝑧) = (0_g‘𝑌)) → (𝑈‘(𝐺‘𝑧)) = (0_g‘𝐴))
86	85	oveq2d 7427	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑧 ∈ ℕ₀) ∧ (𝐺‘𝑧) = (0_g‘𝑌)) → ((𝑧 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑧))) = ((𝑧 ↑ 𝑀)(._r‘𝐴)(0_g‘𝐴)))
87	18	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑧 ∈ ℕ₀) → 𝐴 ∈ Ring)
88	25	ad3antrrr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑧 ∈ ℕ₀) → (mulGrp‘𝐴) ∈ Mnd)
89		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑧 ∈ ℕ₀) → 𝑧 ∈ ℕ₀)
90	28	adantr 481	. . . . . . . . . . . . . . . . . . 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 18977	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑧 ∈ ℕ₀) → (𝑧 ↑ 𝑀) ∈ 𝐵)
92	87, 91	jca 512	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑧 ∈ ℕ₀) → (𝐴 ∈ Ring ∧ (𝑧 ↑ 𝑀) ∈ 𝐵))
93	92	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑧 ∈ ℕ₀) ∧ (𝐺‘𝑧) = (0_g‘𝑌)) → (𝐴 ∈ Ring ∧ (𝑧 ↑ 𝑀) ∈ 𝐵))
94	7, 49, 8	ringrz 20110	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Ring ∧ (𝑧 ↑ 𝑀) ∈ 𝐵) → ((𝑧 ↑ 𝑀)(._r‘𝐴)(0_g‘𝐴)) = (0_g‘𝐴))
95	93, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑧 ∈ ℕ₀) ∧ (𝐺‘𝑧) = (0_g‘𝑌)) → ((𝑧 ↑ 𝑀)(._r‘𝐴)(0_g‘𝐴)) = (0_g‘𝐴))
96	77, 86, 95	3eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑧 ∈ ℕ₀) ∧ (𝐺‘𝑧) = (0_g‘𝑌)) → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0_g‘𝐴))
97	96	ex 413	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑧 ∈ ℕ₀) → ((𝐺‘𝑧) = (0_g‘𝑌) → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0_g‘𝐴)))
98	97	adantlr 713	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑤 ∈ ℕ₀) ∧ 𝑧 ∈ ℕ₀) → ((𝐺‘𝑧) = (0_g‘𝑌) → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0_g‘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) = (0_g‘𝑌)) → (𝑤 < 𝑧 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0_g‘𝐴))))
100	99	ralimdva 3167	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑤 ∈ ℕ₀) → (∀𝑧 ∈ ℕ₀ (𝑤 < 𝑧 → (𝐺‘𝑧) = (0_g‘𝑌)) → ∀𝑧 ∈ ℕ₀ (𝑤 < 𝑧 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0_g‘𝐴))))
101	100	reximdva 3168	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (∃𝑤 ∈ ℕ₀ ∀𝑧 ∈ ℕ₀ (𝑤 < 𝑧 → (𝐺‘𝑧) = (0_g‘𝑌)) → ∃𝑤 ∈ ℕ₀ ∀𝑧 ∈ ℕ₀ (𝑤 < 𝑧 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0_g‘𝐴))))
102	64, 101	syld 47	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝐺 finSupp (0_g‘𝑌) → ∃𝑤 ∈ ℕ₀ ∀𝑧 ∈ ℕ₀ (𝑤 < 𝑧 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0_g‘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))) = (0_g‘𝐴)))
104	53, 54, 103	mptnn0fsupp 13964	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛)))) finSupp (0_g‘𝐴))
105	7, 8, 15, 17, 52, 104	gsumcl 19785	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))))) ∈ 𝐵)
106	32, 9, 7, 43	m2cpminvid 22262	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))))) ∈ 𝐵) → (𝑈‘(𝑇‘(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))))))) = (𝐴 Σ_g (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 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))))))) = (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))))))
108	38, 39	pmatring 22201	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
109	2, 5, 108	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
110		ringmnd 20068	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
111	109, 110	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Mnd)
112	111	ad2antrr 724	. . . . . . 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 22239	. . . . . . . . 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 19104	. . . . . . . 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 728	. . . . . . . 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 728	. . . . . . . . 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 7087	. . . . . . . 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 19810	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ (𝑇‘((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) = (𝑇‘(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛)))))))
125	43, 9, 7, 38, 39, 113	mat2pmatrhm 22243	. . . . . . . . . . . 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 728	. . . . . . . . . 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 20268	. . . . . . . . . 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 22242	. . . . . . . . . . . . 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 728	. . . . . . . . . . 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 18997	. . . . . . . . . . 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 728	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑛 ∈ ℕ₀) → 𝑁 ∈ Fin)
137	5	ad3antrrr 728	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑛 ∈ ℕ₀) → 𝑅 ∈ Ring)
138	31, 32, 43	m2cpminvid2 22264	. . . . . . . . . . 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 7429	. . . . . . . . 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 2772	. . . . . . . 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 5248	. . . . . . 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 7427	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ (𝑇‘((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) = (𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))
144	124, 143	eqtr3d 2774	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑇‘(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) = (𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))
145	144	fveq2d 6895	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈‘(𝑇‘(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))))))) = (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))
146	107, 145	eqtr3d 2774	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))))) = (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))
147	1, 146	sylan9eqr 2794	. 2 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σ_g (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 = (1_r‘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 22396	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀)(._r‘𝐴)(𝑈‘(𝐺‘𝑛))))))
153	147, 152	reximddv2 3212	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ⦋csb 3893 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ↑_m cmap 8822 Fincfn 8941 finSupp cfsupp 9363 0cc0 11112 1c1 11113 + caddc 11115 < clt 11250 − cmin 11446 ℕcn 12214 ℕ₀cn0 12474 ...cfz 13486 Basecbs 17146 ._rcmulr 17200 ·_𝑠 cvsca 17203 0_gc0g 17387 Σ_g cgsu 17388 Mndcmnd 18627 MndHom cmhm 18671 -_gcsg 18823 ._gcmg 18952 GrpHom cghm 19091 CMndccmn 19650 mulGrpcmgp 19989 1_rcur 20006 Ringcrg 20058 CRingccrg 20059 RingHom crh 20252 Poly₁cpl1 21707 coe₁cco1 21708 Mat cmat 21914 ConstPolyMat ccpmat 22212 matToPolyMat cmat2pmat 22213 cPolyMatToMat ccpmat2mat 22214 CharPlyMat cchpmat 22335

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-cur 8254 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-xnn0 12547 df-z 12561 df-dec 12680 df-uz 12825 df-rp 12977 df-fz 13487 df-fzo 13630 df-seq 13969 df-exp 14030 df-hash 14293 df-word 14467 df-lsw 14515 df-concat 14523 df-s1 14548 df-substr 14593 df-pfx 14623 df-splice 14702 df-reverse 14711 df-s2 14801 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-0g 17389 df-gsum 17390 df-prds 17395 df-pws 17397 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-mhm 18673 df-submnd 18674 df-efmnd 18752 df-grp 18824 df-minusg 18825 df-sbg 18826 df-mulg 18953 df-subg 19005 df-ghm 19092 df-gim 19135 df-cntz 19183 df-oppg 19212 df-symg 19237 df-pmtr 19312 df-psgn 19361 df-evpm 19362 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-srg 20012 df-ring 20060 df-cring 20061 df-oppr 20154 df-dvdsr 20175 df-unit 20176 df-invr 20206 df-dvr 20219 df-rnghom 20255 df-subrg 20321 df-drng 20363 df-lmod 20477 df-lss 20548 df-sra 20791 df-rgmod 20792 df-cnfld 20951 df-zring 21024 df-zrh 21059 df-dsmm 21293 df-frlm 21308 df-assa 21414 df-ascl 21416 df-psr 21468 df-mvr 21469 df-mpl 21470 df-opsr 21472 df-psr1 21710 df-vr1 21711 df-ply1 21712 df-coe1 21713 df-mamu 21893 df-mat 21915 df-mdet 22094 df-madu 22143 df-cpmat 22215 df-mat2pmat 22216 df-cpmat2mat 22217 df-decpmat 22272 df-pm2mp 22302 df-chpmat 22336

This theorem is referenced by: cayleyhamilton0 22398 cayleyhamiltonALT 22400

W3C validator