Theorem cpmidpmat 22879

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 22875	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐻 = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(((coe1‘𝐾)‘𝑛) ∗ 𝑂))))))
17	16	fveq2d 6910	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼‘𝐻) = (𝐼‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(((coe1‘𝐾)‘𝑛) ∗ 𝑂)))))))
18		eqid 2737	. . . . . . . . . . 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 22876	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) = (((coe1‘𝐾)‘𝑛) ∗ 𝑂))
20	19	eqcomd 2743	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (((coe1‘𝐾)‘𝑛) ∗ 𝑂) = ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛))
21	20	adantl 481	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (((coe1‘𝐾)‘𝑛) ∗ 𝑂) = ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛))
22	21	fveq2d 6910	. . . . . . 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 7447	. . . . . 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 5242	. . . . 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 7447	. . . 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 6910	. . 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 1149	. . . . 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 22877	. . . . 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 22878	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) finSupp (0g‘𝐴))
30		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → ((coe1‘𝐾)‘𝑘) = ((coe1‘𝐾)‘𝑥))
31	30	oveq1d 7446	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (((coe1‘𝐾)‘𝑘) ∗ 𝑂) = (((coe1‘𝐾)‘𝑥) ∗ 𝑂))
32	31	cbvmptv 5255	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) = (𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))
33	32	eleq1i 2832	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) ∈ (𝐵 ↑m ℕ0) ↔ (𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂)) ∈ (𝐵 ↑m ℕ0))
34	32	breq1i 5150	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) finSupp (0g‘𝐴) ↔ (𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂)) finSupp (0g‘𝐴))
35	33, 34	anbi12i 628	. . . . . 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 22831	. . . . . 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 837	. . . 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 6907	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) = ((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛)
46	45	fveq2i 6909	. . . . . . . 8 ⊢ (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛)) = (𝑇‘((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛))
47	46	oveq2i 7442	. . . . . . 7 ⊢ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛))) = ((𝑛 ↑ 𝑋) · (𝑇‘((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛)))
48	47	mpteq2i 5247	. . . . . 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 7442	. . . . 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 6909	. . . 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 7441	. . . . . 6 ⊢ (((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍)) = (((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍))
52	51	mpteq2i 5247	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍))) = (𝑛 ∈ ℕ0 ↦ (((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍)))
53	52	oveq2i 7442	. . . 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 2802	. . 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 2777	. 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 481	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) = (((coe1‘𝐾)‘𝑛) ∗ 𝑂))
57	56	oveq1d 7446	. . . 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 5242	. . 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 7447	. 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 2781	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼‘𝐻) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 𝑂) ∙ (𝑛𝐸𝑍)))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   class class class wbr 5143   ↦ cmpt 5225  ‘cfv 6561  (class class class)co 7431   ↑_m cmap 8866  Fincfn 8985   finSupp cfsupp 9401  ℕ₀cn0 12526  Basecbs 17247   ·_𝑠 cvsca 17301  0_gc0g 17484   Σ_g cgsu 17485  ._gcmg 19085  mulGrpcmgp 20137  1_rcur 20178  CRingccrg 20231  algSccascl 21872  var₁cv1 22177  Poly₁cpl1 22178  coe₁cco1 22179   Mat cmat 22411   matToPolyMat cmat2pmat 22710   pMatToMatPoly cpm2mp 22798   CharPlyMat cchpmat 22832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-addf 11234  ax-mulf 11235

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1512  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  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 12600  df-z 12614  df-dec 12734  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-word 14553  df-lsw 14601  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-splice 14788  df-reverse 14797  df-s2 14887  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-efmnd 18882  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-gim 19277  df-cntz 19335  df-oppg 19364  df-symg 19387  df-pmtr 19460  df-psgn 19509  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-dvr 20401  df-rhm 20472  df-subrng 20546  df-subrg 20570  df-drng 20731  df-lmod 20860  df-lss 20930  df-sra 21172  df-rgmod 21173  df-cnfld 21365  df-zring 21458  df-zrh 21514  df-dsmm 21752  df-frlm 21767  df-assa 21873  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-mamu 22395  df-mat 22412  df-mdet 22591  df-mat2pmat 22713  df-decpmat 22769  df-pm2mp 22799  df-chpmat 22833

This theorem is referenced by:  chcoeffeq  22892