Theorem cpmadumatpoly 22879

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

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 2726	. . 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 2726	. . 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 2730	. . . . . 6 ⊢ (𝑛 = 𝑧 → (𝑛 = 0 ↔ 𝑧 = 0))
18		eqeq1 2730	. . . . . . 7 ⊢ (𝑛 = 𝑧 → (𝑛 = (𝑠 + 1) ↔ 𝑧 = (𝑠 + 1)))
19		breq2 5159	. . . . . . . 8 ⊢ (𝑛 = 𝑧 → ((𝑠 + 1) < 𝑛 ↔ (𝑠 + 1) < 𝑧))
20		fvoveq1 7449	. . . . . . . . . 10 ⊢ (𝑛 = 𝑧 → (𝑏‘(𝑛 − 1)) = (𝑏‘(𝑧 − 1)))
21	20	fveq2d 6907	. . . . . . . . 9 ⊢ (𝑛 = 𝑧 → (𝑇‘(𝑏‘(𝑛 − 1))) = (𝑇‘(𝑏‘(𝑧 − 1))))
22		2fveq3 6908	. . . . . . . . . 10 ⊢ (𝑛 = 𝑧 → (𝑇‘(𝑏‘𝑛)) = (𝑇‘(𝑏‘𝑧)))
23	22	oveq2d 7442	. . . . . . . . 9 ⊢ (𝑛 = 𝑧 → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑧))))
24	21, 23	oveq12d 7444	. . . . . . . 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 4559	. . . . . . 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 4559	. . . . . 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 4559	. . . . 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 5268	. . . 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 2754	. . 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 22874	. 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 1133	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
32	31	ad3antrrr 728	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
33		crngring 20230	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
34	33	3ad2ant2 1131	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
35	34	ad3antrrr 728	. . . . . . . . . . 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 22851	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶𝑆)
38	33, 37	syl3anl2 1410	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶𝑆)
39	38	anassrs 466	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐺:ℕ0⟶𝑆)
40	39	ffvelcdmda 7100	. . . . . . . . . . 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 22751	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐺‘𝑛) ∈ 𝑆) → (𝑇‘(𝑈‘(𝐺‘𝑛))) = (𝐺‘𝑛))
43	32, 35, 40, 42	syl3anc 1368	. . . . . . . . . 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 2732	. . . . . . . . 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 7442	. . . . . . . 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 5255	. . . . . . 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 7442	. . . . . 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 2737	. . . . 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 6903	. . . . . . 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 1145	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing))
51	50	ad2antrr 724	. . . . . . . . 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 22877	. . . . . . . . 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 22878	. . . . . . . . 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 22821	. . . . . . . . 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 835	. . . . . . . 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 7003	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:ℕ0⟶𝑆 ∧ 𝑛 ∈ ℕ0) → ((𝑈 ∘ 𝐺)‘𝑛) = (𝑈‘(𝐺‘𝑛)))
63	62	eqcomd 2732	. . . . . . . . . . . . . 14 ⊢ ((𝐺:ℕ0⟶𝑆 ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺‘𝑛)) = ((𝑈 ∘ 𝐺)‘𝑛))
64	39, 63	sylan 578	. . . . . . . . . . . . 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 6907	. . . . . . . . . . . 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 7442	. . . . . . . . . . 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 5255	. . . . . . . . . 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 7442	. . . . . . . . 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 6907	. . . . . . . 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 7441	. . . . . . . . . 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 5255	. . . . . . . . 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 7442	. . . . . . . 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 2776	. . . . . . 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 2788	. . . . . 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 411	. . . . 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 239	. . . 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 3158	. . 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 3158	. 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 15	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 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  ∃wrex 3060  ifcif 4533   class class class wbr 5155   ↦ cmpt 5238   ∘ ccom 5688  ⟶wf 6552  ‘cfv 6556  (class class class)co 7426   ↑_m cmap 8857  Fincfn 8976   finSupp cfsupp 9407  0cc0 11160  1c1 11161   + caddc 11163   < clt 11300   − cmin 11496  ℕcn 12266  ℕ₀cn0 12526  ...cfz 13540  Basecbs 17215  +_gcplusg 17268  ._rcmulr 17269   ·_𝑠 cvsca 17272  0_gc0g 17456   Σ_g cgsu 17457  -_gcsg 18932  ._gcmg 19063  mulGrpcmgp 20119  1_rcur 20166  Ringcrg 20218  CRingccrg 20219  var₁cv1 22167  Poly₁cpl1 22168   Mat cmat 22401   maAdju cmadu 22628   ConstPolyMat ccpmat 22699   matToPolyMat cmat2pmat 22700   cPolyMatToMat ccpmat2mat 22701   pMatToMatPoly cpm2mp 22788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5292  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748  ax-cnex 11216  ax-resscn 11217  ax-1cn 11218  ax-icn 11219  ax-addcl 11220  ax-addrcl 11221  ax-mulcl 11222  ax-mulrcl 11223  ax-mulcom 11224  ax-addass 11225  ax-mulass 11226  ax-distr 11227  ax-i2m1 11228  ax-1ne0 11229  ax-1rid 11230  ax-rnegex 11231  ax-rrecex 11232  ax-cnre 11233  ax-pre-lttri 11234  ax-pre-lttrn 11235  ax-pre-ltadd 11236  ax-pre-mulgt0 11237  ax-addf 11239  ax-mulf 11240

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-xor 1506  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-ot 4642  df-uni 4916  df-int 4957  df-iun 5005  df-iin 5006  df-br 5156  df-opab 5218  df-mpt 5239  df-tr 5273  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5639  df-se 5640  df-we 5641  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6314  df-ord 6381  df-on 6382  df-lim 6383  df-suc 6384  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-isom 6565  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7692  df-ofr 7693  df-om 7879  df-1st 8005  df-2nd 8006  df-supp 8177  df-tpos 8243  df-cur 8284  df-frecs 8298  df-wrecs 8329  df-recs 8403  df-rdg 8442  df-1o 8498  df-2o 8499  df-er 8736  df-map 8859  df-pm 8860  df-ixp 8929  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-fsupp 9408  df-sup 9487  df-oi 9555  df-card 9984  df-pnf 11302  df-mnf 11303  df-xr 11304  df-ltxr 11305  df-le 11306  df-sub 11498  df-neg 11499  df-div 11924  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12599  df-z 12613  df-dec 12732  df-uz 12877  df-rp 13031  df-fz 13541  df-fzo 13684  df-seq 14024  df-exp 14084  df-hash 14350  df-word 14525  df-lsw 14573  df-concat 14581  df-s1 14606  df-substr 14651  df-pfx 14681  df-splice 14760  df-reverse 14769  df-s2 14859  df-struct 17151  df-sets 17168  df-slot 17186  df-ndx 17198  df-base 17216  df-ress 17245  df-plusg 17281  df-mulr 17282  df-starv 17283  df-sca 17284  df-vsca 17285  df-ip 17286  df-tset 17287  df-ple 17288  df-ds 17290  df-unif 17291  df-hom 17292  df-cco 17293  df-0g 17458  df-gsum 17459  df-prds 17464  df-pws 17466  df-mre 17601  df-mrc 17602  df-acs 17604  df-mgm 18635  df-sgrp 18714  df-mnd 18730  df-mhm 18775  df-submnd 18776  df-efmnd 18861  df-grp 18933  df-minusg 18934  df-sbg 18935  df-mulg 19064  df-subg 19119  df-ghm 19209  df-gim 19255  df-cntz 19313  df-oppg 19342  df-symg 19367  df-pmtr 19442  df-psgn 19491  df-cmn 19782  df-abl 19783  df-mgp 20120  df-rng 20138  df-ur 20167  df-srg 20172  df-ring 20220  df-cring 20221  df-oppr 20318  df-dvdsr 20341  df-unit 20342  df-invr 20372  df-dvr 20385  df-rhm 20456  df-subrng 20530  df-subrg 20555  df-drng 20711  df-lmod 20840  df-lss 20911  df-sra 21153  df-rgmod 21154  df-cnfld 21346  df-zring 21439  df-zrh 21495  df-dsmm 21732  df-frlm 21747  df-assa 21853  df-ascl 21855  df-psr 21908  df-mvr 21909  df-mpl 21910  df-opsr 21912  df-psr1 22171  df-vr1 22172  df-ply1 22173  df-coe1 22174  df-mamu 22385  df-mat 22402  df-mdet 22581  df-madu 22630  df-cpmat 22702  df-mat2pmat 22703  df-cpmat2mat 22704  df-decpmat 22759  df-pm2mp 22789

This theorem is referenced by:  chcoeffeq  22882