Theorem cpmidpmat 22022

Description: Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as polynomial over the ring of matrices. (Contributed by AV, 14-Nov-2019.) (Revised by AV, 7-Dec-2019.)

Hypotheses

Ref	Expression
cpmidgsum.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cpmidgsum.b	⊢ 𝐵 = (Base‘𝐴)
cpmidgsum.p	⊢ 𝑃 = (Poly1‘𝑅)
cpmidgsum.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
cpmidgsum.x	⊢ 𝑋 = (var1‘𝑅)
cpmidgsum.e	⊢ ↑ = (.g‘(mulGrp‘𝑃))
cpmidgsum.m	⊢ · = ( ·𝑠 ‘𝑌)
cpmidgsum.1	⊢ 1 = (1r‘𝑌)
cpmidgsum.u	⊢ 𝑈 = (algSc‘𝑃)
cpmidgsum.c	⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
cpmidgsum.k	⊢ 𝐾 = (𝐶‘𝑀)
cpmidgsum.h	⊢ 𝐻 = (𝐾 · 1 )
cpmidgsumm2pm.o	⊢ 𝑂 = (1r‘𝐴)
cpmidgsumm2pm.m	⊢ ∗ = ( ·𝑠 ‘𝐴)
cpmidgsumm2pm.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
cpmidgsum.w	⊢ 𝑊 = (Base‘𝑌)
cpmidpmat.p	⊢ 𝑄 = (Poly1‘𝐴)
cpmidpmat.z	⊢ 𝑍 = (var1‘𝐴)
cpmidpmat.m	⊢ ∙ = ( ·𝑠 ‘𝑄)
cpmidpmat.e	⊢ 𝐸 = (.g‘(mulGrp‘𝑄))
cpmidpmat.i	⊢ 𝐼 = (𝑁 pMatToMatPoly 𝑅)

Assertion

Ref	Expression
cpmidpmat	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼‘𝐻) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 𝑂) ∙ (𝑛𝐸𝑍)))))

Distinct variable groups:   𝐴,𝑛   𝐵,𝑛   𝑛,𝐻   𝑛,𝐾   𝑛,𝑋   𝑛,𝑁   𝑃,𝑛   𝑅,𝑛   𝑛,𝑌   ↑ ,𝑛   𝑛,𝑀   𝑛,𝐼   𝑛,𝑂   𝑇,𝑛   𝑛,𝑊   ∗ ,𝑛   · ,𝑛

Allowed substitution hints:   𝐶(𝑛)   𝑄(𝑛)   ∙ (𝑛)   𝑈(𝑛)   1 (𝑛)   𝐸(𝑛)   𝑍(𝑛)

Proof of Theorem cpmidpmat

Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cpmidgsum.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		cpmidgsum.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
3		cpmidgsum.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
4		cpmidgsum.y	. . . 4 ⊢ 𝑌 = (𝑁 Mat 𝑃)
5		cpmidgsum.x	. . . 4 ⊢ 𝑋 = (var1‘𝑅)
6		cpmidgsum.e	. . . 4 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
7		cpmidgsum.m	. . . 4 ⊢ · = ( ·𝑠 ‘𝑌)
8		cpmidgsum.1	. . . 4 ⊢ 1 = (1r‘𝑌)
9		cpmidgsum.u	. . . 4 ⊢ 𝑈 = (algSc‘𝑃)
10		cpmidgsum.c	. . . 4 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
11		cpmidgsum.k	. . . 4 ⊢ 𝐾 = (𝐶‘𝑀)
12		cpmidgsum.h	. . . 4 ⊢ 𝐻 = (𝐾 · 1 )
13		cpmidgsumm2pm.o	. . . 4 ⊢ 𝑂 = (1r‘𝐴)
14		cpmidgsumm2pm.m	. . . 4 ⊢ ∗ = ( ·𝑠 ‘𝐴)
15		cpmidgsumm2pm.t	. . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	cpmidgsumm2pm 22018	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐻 = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(((coe1‘𝐾)‘𝑛) ∗ 𝑂))))))
17	16	fveq2d 6778	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼‘𝐻) = (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(((coe1‘𝐾)‘𝑛) ∗ 𝑂)))))))
18		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) = (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18	cpmidpmatlem1 22019	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) = (((coe1‘𝐾)‘𝑛) ∗ 𝑂))
20	19	eqcomd 2744	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (((coe1‘𝐾)‘𝑛) ∗ 𝑂) = ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛))
21	20	adantl 482	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (((coe1‘𝐾)‘𝑛) ∗ 𝑂) = ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛))
22	21	fveq2d 6778	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(((coe1‘𝐾)‘𝑛) ∗ 𝑂)) = (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛)))
23	22	oveq2d 7291	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑋) · (𝑇‘(((coe1‘𝐾)‘𝑛) ∗ 𝑂))) = ((𝑛 ↑ 𝑋) · (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛))))
24	23	mpteq2dva 5174	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(((coe1‘𝐾)‘𝑛) ∗ 𝑂)))) = (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛)))))
25	24	oveq2d 7291	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(((coe1‘𝐾)‘𝑛) ∗ 𝑂))))) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛))))))
26	25	fveq2d 6778	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(((coe1‘𝐾)‘𝑛) ∗ 𝑂)))))) = (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛)))))))
27		3simpa 1147	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing))
28	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18	cpmidpmatlem2 22020	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) ∈ (𝐵 ↑m ℕ0))
29	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18	cpmidpmatlem3 22021	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) finSupp (0g‘𝐴))
30		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → ((coe1‘𝐾)‘𝑘) = ((coe1‘𝐾)‘𝑥))
31	30	oveq1d 7290	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (((coe1‘𝐾)‘𝑘) ∗ 𝑂) = (((coe1‘𝐾)‘𝑥) ∗ 𝑂))
32	31	cbvmptv 5187	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) = (𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))
33	32	eleq1i 2829	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) ∈ (𝐵 ↑m ℕ0) ↔ (𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂)) ∈ (𝐵 ↑m ℕ0))
34	32	breq1i 5081	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) finSupp (0g‘𝐴) ↔ (𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂)) finSupp (0g‘𝐴))
35	33, 34	anbi12i 627	. . . . . 6 ⊢ (((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) ∈ (𝐵 ↑m ℕ0) ∧ (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) finSupp (0g‘𝐴)) ↔ ((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂)) ∈ (𝐵 ↑m ℕ0) ∧ (𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂)) finSupp (0g‘𝐴)))
36		cpmidgsum.w	. . . . . . 7 ⊢ 𝑊 = (Base‘𝑌)
37		cpmidpmat.m	. . . . . . 7 ⊢ ∙ = ( ·𝑠 ‘𝑄)
38		cpmidpmat.e	. . . . . . 7 ⊢ 𝐸 = (.g‘(mulGrp‘𝑄))
39		cpmidpmat.z	. . . . . . 7 ⊢ 𝑍 = (var1‘𝐴)
40		cpmidpmat.p	. . . . . . 7 ⊢ 𝑄 = (Poly1‘𝐴)
41		cpmidpmat.i	. . . . . . 7 ⊢ 𝐼 = (𝑁 pMatToMatPoly 𝑅)
42	3, 4, 36, 37, 38, 39, 1, 2, 40, 41, 6, 5, 7, 15	pm2mp 21974	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ ((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂)) ∈ (𝐵 ↑m ℕ0) ∧ (𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂)) finSupp (0g‘𝐴))) → (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛)))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍)))))
43	35, 42	sylan2b 594	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) ∈ (𝐵 ↑m ℕ0) ∧ (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) finSupp (0g‘𝐴))) → (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛)))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍)))))
44	27, 28, 29, 43	syl12anc 834	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛)))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍)))))
45	32	fveq1i 6775	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) = ((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛)
46	45	fveq2i 6777	. . . . . . . 8 ⊢ (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛)) = (𝑇‘((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛))
47	46	oveq2i 7286	. . . . . . 7 ⊢ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛))) = ((𝑛 ↑ 𝑋) · (𝑇‘((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛)))
48	47	mpteq2i 5179	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛)))) = (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛))))
49	48	oveq2i 7286	. . . . 5 ⊢ (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛))))) = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛)))))
50	49	fveq2i 6777	. . . 4 ⊢ (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛)))))) = (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛))))))
51	45	oveq1i 7285	. . . . . 6 ⊢ (((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍)) = (((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍))
52	51	mpteq2i 5179	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍))) = (𝑛 ∈ ℕ0 ↦ (((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍)))
53	52	oveq2i 7286	. . . 4 ⊢ (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍)))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍))))
54	44, 50, 53	3eqtr4g 2803	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛)))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍)))))
55	26, 54	eqtrd 2778	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(((coe1‘𝐾)‘𝑛) ∗ 𝑂)))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍)))))
56	19	adantl 482	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) = (((coe1‘𝐾)‘𝑛) ∗ 𝑂))
57	56	oveq1d 7290	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍)) = ((((coe1‘𝐾)‘𝑛) ∗ 𝑂) ∙ (𝑛𝐸𝑍)))
58	57	mpteq2dva 5174	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑛 ∈ ℕ0 ↦ (((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍))) = (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 𝑂) ∙ (𝑛𝐸𝑍))))
59	58	oveq2d 7291	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍)))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 𝑂) ∙ (𝑛𝐸𝑍)))))
60	17, 55, 59	3eqtrd 2782	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼‘𝐻) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 𝑂) ∙ (𝑛𝐸𝑍)))))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   class class class wbr 5074   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  Fincfn 8733   finSupp cfsupp 9128  ℕ₀cn0 12233  Basecbs 16912   ·_𝑠 cvsca 16966  0_gc0g 17150   Σ_g cgsu 17151  ._gcmg 18700  mulGrpcmgp 19720  1_rcur 19737  CRingccrg 19784  algSccascl 21059  var₁cv1 21347  Poly₁cpl1 21348  coe₁cco1 21349   Mat cmat 21554   matToPolyMat cmat2pmat 21853   pMatToMatPoly cpm2mp 21941   CharPlyMat cchpmat 21975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-addf 10950  ax-mulf 10951

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-xor 1507  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-ot 4570  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-ofr 7534  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-tpos 8042  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-sup 9201  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-xnn0 12306  df-z 12320  df-dec 12438  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-word 14218  df-lsw 14266  df-concat 14274  df-s1 14301  df-substr 14354  df-pfx 14384  df-splice 14463  df-reverse 14472  df-s2 14561  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-0g 17152  df-gsum 17153  df-prds 17158  df-pws 17160  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-efmnd 18508  df-grp 18580  df-minusg 18581  df-sbg 18582  df-mulg 18701  df-subg 18752  df-ghm 18832  df-gim 18875  df-cntz 18923  df-oppg 18950  df-symg 18975  df-pmtr 19050  df-psgn 19099  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-srg 19742  df-ring 19785  df-cring 19786  df-oppr 19862  df-dvdsr 19883  df-unit 19884  df-invr 19914  df-dvr 19925  df-rnghom 19959  df-drng 19993  df-subrg 20022  df-lmod 20125  df-lss 20194  df-sra 20434  df-rgmod 20435  df-cnfld 20598  df-zring 20671  df-zrh 20705  df-dsmm 20939  df-frlm 20954  df-assa 21060  df-ascl 21062  df-psr 21112  df-mvr 21113  df-mpl 21114  df-opsr 21116  df-psr1 21351  df-vr1 21352  df-ply1 21353  df-coe1 21354  df-mamu 21533  df-mat 21555  df-mdet 21734  df-mat2pmat 21856  df-decpmat 21912  df-pm2mp 21942  df-chpmat 21976

This theorem is referenced by:  chcoeffeq  22035