Theorem cpmadumatpoly 23021

Description: The product of the characteristic matrix of a given matrix and its adjunct represented as a polynomial over matrices. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 7-Dec-2019.)

Hypotheses

Ref	Expression
cpmadumatpoly.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cpmadumatpoly.b	⊢ 𝐵 = (Base‘𝐴)
cpmadumatpoly.p	⊢ 𝑃 = (Poly1‘𝑅)
cpmadumatpoly.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
cpmadumatpoly.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
cpmadumatpoly.r	⊢ × = (.r‘𝑌)
cpmadumatpoly.m0	⊢ − = (-g‘𝑌)
cpmadumatpoly.0	⊢ 0 = (0g‘𝑌)
cpmadumatpoly.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))))))))
cpmadumatpoly.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
cpmadumatpoly.m1	⊢ · = ( ·𝑠 ‘𝑌)
cpmadumatpoly.1	⊢ 1 = (1r‘𝑌)
cpmadumatpoly.z	⊢ 𝑍 = (var1‘𝑅)
cpmadumatpoly.d	⊢ 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀))
cpmadumatpoly.j	⊢ 𝐽 = (𝑁 maAdju 𝑃)
cpmadumatpoly.w	⊢ 𝑊 = (Base‘𝑌)
cpmadumatpoly.q	⊢ 𝑄 = (Poly1‘𝐴)
cpmadumatpoly.x	⊢ 𝑋 = (var1‘𝐴)
cpmadumatpoly.m2	⊢ ∗ = ( ·𝑠 ‘𝑄)
cpmadumatpoly.e	⊢ ↑ = (.g‘(mulGrp‘𝑄))
cpmadumatpoly.u	⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
cpmadumatpoly.i	⊢ 𝐼 = (𝑁 pMatToMatPoly 𝑅)

Assertion

Ref	Expression
cpmadumatpoly	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐼‘(𝐷 × (𝐽‘𝐷))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋)))))

Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑆,𝑛   𝑛,𝑌,𝑏,𝑠   − ,𝑛   𝐷,𝑏,𝑛,𝑠   𝐴,𝑏,𝑛,𝑠   1 ,𝑛   𝑁,𝑏,𝑠   𝑛,𝐺   𝑛,𝑊   𝑅,𝑏,𝑠   · ,𝑏,𝑛,𝑠   𝑇,𝑏,𝑛,𝑠   𝑍,𝑏,𝑛,𝑠   𝑈,𝑛   𝐵,𝑏,𝑠   𝑀,𝑏,𝑠   𝑌,𝑏,𝑠   𝑛,𝐼   × ,𝑛   𝐽,𝑏,𝑛,𝑠   0 ,𝑛   𝑃,𝑏,𝑛,𝑠

Allowed substitution hints:   𝑄(𝑛,𝑠,𝑏)   𝑆(𝑠,𝑏)   × (𝑠,𝑏)   𝑈(𝑠,𝑏)   1 (𝑠,𝑏)   ↑ (𝑛,𝑠,𝑏)   𝐺(𝑠,𝑏)   𝐼(𝑠,𝑏)   ∗ (𝑛,𝑠,𝑏)   − (𝑠,𝑏)   𝑊(𝑠,𝑏)   𝑋(𝑛,𝑠,𝑏)   0 (𝑠,𝑏)

Proof of Theorem cpmadumatpoly

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cpmadumatpoly.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		cpmadumatpoly.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
3		cpmadumatpoly.p	. . 3 ⊢ 𝑃 = (Poly1‘𝑅)
4		cpmadumatpoly.y	. . 3 ⊢ 𝑌 = (𝑁 Mat 𝑃)
5		cpmadumatpoly.t	. . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
6		cpmadumatpoly.z	. . 3 ⊢ 𝑍 = (var1‘𝑅)
7		eqid 2763	. . 3 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
8		cpmadumatpoly.m1	. . 3 ⊢ · = ( ·𝑠 ‘𝑌)
9		cpmadumatpoly.r	. . 3 ⊢ × = (.r‘𝑌)
10		cpmadumatpoly.1	. . 3 ⊢ 1 = (1r‘𝑌)
11		eqid 2763	. . 3 ⊢ (+g‘𝑌) = (+g‘𝑌)
12		cpmadumatpoly.m0	. . 3 ⊢ − = (-g‘𝑌)
13		cpmadumatpoly.d	. . 3 ⊢ 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀))
14		cpmadumatpoly.j	. . 3 ⊢ 𝐽 = (𝑁 maAdju 𝑃)
15		cpmadumatpoly.0	. . 3 ⊢ 0 = (0g‘𝑌)
16		cpmadumatpoly.g	. . . 4 ⊢ 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))))))))
17		eqeq1 2767	. . . . . 6 ⊢ (𝑛 = 𝑧 → (𝑛 = 0 ↔ 𝑧 = 0))
18		eqeq1 2767	. . . . . . 7 ⊢ (𝑛 = 𝑧 → (𝑛 = (𝑠 + 1) ↔ 𝑧 = (𝑠 + 1)))
19		breq2 5114	. . . . . . . 8 ⊢ (𝑛 = 𝑧 → ((𝑠 + 1) < 𝑛 ↔ (𝑠 + 1) < 𝑧))
20		fvoveq1 7435	. . . . . . . . . 10 ⊢ (𝑛 = 𝑧 → (𝑏‘(𝑛 − 1)) = (𝑏‘(𝑧 − 1)))
21	20	fveq2d 6887	. . . . . . . . 9 ⊢ (𝑛 = 𝑧 → (𝑇‘(𝑏‘(𝑛 − 1))) = (𝑇‘(𝑏‘(𝑧 − 1))))
22		2fveq3 6888	. . . . . . . . . 10 ⊢ (𝑛 = 𝑧 → (𝑇‘(𝑏‘𝑛)) = (𝑇‘(𝑏‘𝑧)))
23	22	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑛 = 𝑧 → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑧))))
24	21, 23	oveq12d 7430	. . . . . . . 8 ⊢ (𝑛 = 𝑧 → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) = ((𝑇‘(𝑏‘(𝑧 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑧)))))
25	19, 24	ifbieq2d 4515	. . . . . . 7 ⊢ (𝑛 = 𝑧 → if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))) = if((𝑠 + 1) < 𝑧, 0 , ((𝑇‘(𝑏‘(𝑧 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑧))))))
26	18, 25	ifbieq2d 4515	. . . . . 6 ⊢ (𝑛 = 𝑧 → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = if(𝑧 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑧, 0 , ((𝑇‘(𝑏‘(𝑧 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑧)))))))
27	17, 26	ifbieq2d 4515	. . . . 5 ⊢ (𝑛 = 𝑧 → 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))))))) = if(𝑧 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑧 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑧, 0 , ((𝑇‘(𝑏‘(𝑧 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑧))))))))
28	27	cbvmptv 5216	. . . 4 ⊢ (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)))))))) = (𝑧 ∈ ℕ0 ↦ if(𝑧 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑧 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑧, 0 , ((𝑇‘(𝑏‘(𝑧 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑧))))))))
29	16, 28	eqtri 2786	. . 3 ⊢ 𝐺 = (𝑧 ∈ ℕ0 ↦ if(𝑧 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑧 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑧, 0 , ((𝑇‘(𝑏‘(𝑧 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑧))))))))
30	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 29	cpmadugsum 23016	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐷 × (𝐽‘𝐷)) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝐺‘𝑛)))))
31		simp1 1154	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
32	31	ad3antrrr 742	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
33		crngring 20328	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
34	33	3ad2ant2 1152	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
35	34	ad3antrrr 742	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
36		cpmadumatpoly.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
37	1, 2, 3, 4, 9, 12, 15, 5, 16, 36	chfacfisfcpmat 22993	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶𝑆)
38	33, 37	syl3anl2 1440	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶𝑆)
39	38	anassrs 472	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐺:ℕ0⟶𝑆)
40	39	ffvelcdmda 7081	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝐺‘𝑛) ∈ 𝑆)
41		cpmadumatpoly.u	. . . . . . . . . . . 12 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
42	36, 41, 5	m2cpminvid2 22893	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐺‘𝑛) ∈ 𝑆) → (𝑇‘(𝑈‘(𝐺‘𝑛))) = (𝐺‘𝑛))
43	32, 35, 40, 42	syl3anc 1398	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑈‘(𝐺‘𝑛))) = (𝐺‘𝑛))
44	43	eqcomd 2769	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝐺‘𝑛) = (𝑇‘(𝑈‘(𝐺‘𝑛))))
45	44	oveq2d 7428	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝐺‘𝑛)) = ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘(𝑈‘(𝐺‘𝑛)))))
46	45	mpteq2dva 5205	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝐺‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘(𝑈‘(𝐺‘𝑛))))))
47	46	oveq2d 7428	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝐺‘𝑛)))) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘(𝑈‘(𝐺‘𝑛)))))))
48	47	eqeq2d 2774	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → ((𝐷 × (𝐽‘𝐷)) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝐺‘𝑛)))) ↔ (𝐷 × (𝐽‘𝐷)) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘(𝑈‘(𝐺‘𝑛))))))))
49		fveq2 6883	. . . . . . 7 ⊢ ((𝐷 × (𝐽‘𝐷)) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘(𝑈‘(𝐺‘𝑛)))))) → (𝐼‘(𝐷 × (𝐽‘𝐷))) = (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘(𝑈‘(𝐺‘𝑛))))))))
50		3simpa 1166	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing))
51	50	ad2antrr 738	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing))
52		cpmadumatpoly.w	. . . . . . . . . 10 ⊢ 𝑊 = (Base‘𝑌)
53		cpmadumatpoly.q	. . . . . . . . . 10 ⊢ 𝑄 = (Poly1‘𝐴)
54		cpmadumatpoly.x	. . . . . . . . . 10 ⊢ 𝑋 = (var1‘𝐴)
55		cpmadumatpoly.m2	. . . . . . . . . 10 ⊢ ∗ = ( ·𝑠 ‘𝑄)
56		cpmadumatpoly.e	. . . . . . . . . 10 ⊢ ↑ = (.g‘(mulGrp‘𝑄))
57	1, 2, 3, 4, 5, 9, 12, 15, 16, 36, 8, 10, 6, 13, 14, 52, 53, 54, 55, 56, 41	cpmadumatpolylem1 23019	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0))
58	1, 2, 3, 4, 5, 9, 12, 15, 16, 36, 8, 10, 6, 13, 14, 52, 53, 54, 55, 56, 41	cpmadumatpolylem2 23020	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑈 ∘ 𝐺) finSupp (0g‘𝐴))
59		cpmadumatpoly.i	. . . . . . . . . 10 ⊢ 𝐼 = (𝑁 pMatToMatPoly 𝑅)
60	3, 4, 52, 55, 56, 54, 1, 2, 53, 59, 7, 6, 8, 5	pm2mp 22963	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ ((𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0) ∧ (𝑈 ∘ 𝐺) finSupp (0g‘𝐴))) → (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘((𝑈 ∘ 𝐺)‘𝑛)))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝑈 ∘ 𝐺)‘𝑛) ∗ (𝑛 ↑ 𝑋)))))
61	51, 57, 58, 60	syl12anc 849	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘((𝑈 ∘ 𝐺)‘𝑛)))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝑈 ∘ 𝐺)‘𝑛) ∗ (𝑛 ↑ 𝑋)))))
62		fvco3 6983	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:ℕ0⟶𝑆 ∧ 𝑛 ∈ ℕ0) → ((𝑈 ∘ 𝐺)‘𝑛) = (𝑈‘(𝐺‘𝑛)))
63	62	eqcomd 2769	. . . . . . . . . . . . . 14 ⊢ ((𝐺:ℕ0⟶𝑆 ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺‘𝑛)) = ((𝑈 ∘ 𝐺)‘𝑛))
64	39, 63	sylan 591	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺‘𝑛)) = ((𝑈 ∘ 𝐺)‘𝑛))
65	64	fveq2d 6887	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑈‘(𝐺‘𝑛))) = (𝑇‘((𝑈 ∘ 𝐺)‘𝑛)))
66	65	oveq2d 7428	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘(𝑈‘(𝐺‘𝑛)))) = ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘((𝑈 ∘ 𝐺)‘𝑛))))
67	66	mpteq2dva 5205	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘(𝑈‘(𝐺‘𝑛))))) = (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘((𝑈 ∘ 𝐺)‘𝑛)))))
68	67	oveq2d 7428	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘(𝑈‘(𝐺‘𝑛)))))) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘((𝑈 ∘ 𝐺)‘𝑛))))))
69	68	fveq2d 6887	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘(𝑈‘(𝐺‘𝑛))))))) = (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘((𝑈 ∘ 𝐺)‘𝑛)))))))
70	64	oveq1d 7427	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋)) = (((𝑈 ∘ 𝐺)‘𝑛) ∗ (𝑛 ↑ 𝑋)))
71	70	mpteq2dva 5205	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋))) = (𝑛 ∈ ℕ0 ↦ (((𝑈 ∘ 𝐺)‘𝑛) ∗ (𝑛 ↑ 𝑋))))
72	71	oveq2d 7428	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋)))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝑈 ∘ 𝐺)‘𝑛) ∗ (𝑛 ↑ 𝑋)))))
73	61, 69, 72	3eqtr4d 2808	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘(𝑈‘(𝐺‘𝑛))))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋)))))
74	49, 73	sylan9eqr 2820	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ (𝐷 × (𝐽‘𝐷)) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘(𝑈‘(𝐺‘𝑛))))))) → (𝐼‘(𝐷 × (𝐽‘𝐷))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋)))))
75	74	ex 417	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → ((𝐷 × (𝐽‘𝐷)) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝑇‘(𝑈‘(𝐺‘𝑛)))))) → (𝐼‘(𝐷 × (𝐽‘𝐷))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋))))))
76	48, 75	sylbid 243	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → ((𝐷 × (𝐽‘𝐷)) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝐺‘𝑛)))) → (𝐼‘(𝐷 × (𝐽‘𝐷))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋))))))
77	76	reximdva 3178	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) → (∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐷 × (𝐽‘𝐷)) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝐺‘𝑛)))) → ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐼‘(𝐷 × (𝐽‘𝐷))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋))))))
78	77	reximdva 3178	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐷 × (𝐽‘𝐷)) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘𝑃))𝑍) · (𝐺‘𝑛)))) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐼‘(𝐷 × (𝐽‘𝐷))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋))))))
79	30, 78	mpd 16	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐼‘(𝐷 × (𝐽‘𝐷))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋)))))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  ifcif 4488   class class class wbr 5110   ↦ cmpt 5193   ∘ ccom 5667  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412   ↑_m cmap 8825  Fincfn 8944   finSupp cfsupp 9322  0cc0 11101  1c1 11102   + caddc 11104   < clt 11244   − cmin 11442  ℕcn 12234  ℕ₀cn0 12505  ...cfz 13536  Basecbs 17270  +_gcplusg 17311  ._rcmulr 17312   ·_𝑠 cvsca 17315  0_gc0g 17493   Σ_g cgsu 17494  -_gcsg 19003  ._gcmg 19134  mulGrpcmgp 20217  1_rcur 20264  Ringcrg 20316  CRingccrg 20317  var₁cv1 22317  Poly₁cpl1 22318   Mat cmat 22545   maAdju cmadu 22770   ConstPolyMat ccpmat 22841   matToPolyMat cmat2pmat 22842   cPolyMatToMat ccpmat2mat 22843   pMatToMatPoly cpm2mp 22930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178  ax-addf 11180  ax-mulf 11181

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-ofr 7677  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8158  df-tpos 8223  df-cur 8264  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-er 8695  df-map 8827  df-pm 8828  df-ixp 8897  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-fsupp 9323  df-sup 9403  df-oi 9473  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-div 11873  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-xnn0 12579  df-z 12593  df-dec 12713  df-uz 12864  df-rp 13018  df-fz 13537  df-fzo 13685  df-seq 14040  df-exp 14100  df-hash 14369  df-word 14553  df-lsw 14602  df-concat 14610  df-s1 14636  df-substr 14681  df-pfx 14711  df-splice 14789  df-reverse 14798  df-s2 14887  df-struct 17208  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  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 17495  df-gsum 17496  df-prds 17501  df-pws 17503  df-mre 17639  df-mrc 17640  df-acs 17642  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-mhm 18842  df-submnd 18843  df-efmnd 18929  df-grp 19004  df-minusg 19005  df-sbg 19006  df-mulg 19135  df-subg 19190  df-ghm 19285  df-gim 19330  df-cntz 19388  df-oppg 19417  df-symg 19441  df-pmtr 19513  df-psgn 19562  df-cmn 19853  df-abl 19854  df-mgp 20218  df-rng 20232  df-ur 20265  df-srg 20270  df-ring 20318  df-cring 20319  df-oppr 20420  df-dvdsr 20440  df-unit 20441  df-invr 20471  df-dvr 20484  df-rhm 20555  df-subrng 20632  df-subrg 20656  df-drng 20816  df-lmod 20964  df-lss 21034  df-sra 21275  df-rgmod 21276  df-cnfld 21504  df-zring 21578  df-zrh 21634  df-dsmm 21863  df-frlm 21878  df-assa 21984  df-ascl 21986  df-psr 22040  df-mvr 22041  df-mpl 22042  df-opsr 22044  df-psr1 22321  df-vr1 22322  df-ply1 22323  df-coe1 22324  df-mamu 22529  df-mat 22546  df-mdet 22723  df-madu 22772  df-cpmat 22844  df-mat2pmat 22845  df-cpmat2mat 22846  df-decpmat 22901  df-pm2mp 22931

This theorem is referenced by:  chcoeffeq  23024