Theorem cpmidpmat 21930

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 21926	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐻 = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(((coe1‘𝐾)‘𝑛) ∗ 𝑂))))))
17	16	fveq2d 6760	. 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 21927	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (((coe1‘𝐾)‘𝑛) ∗ 𝑂) = ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛))
22	21	fveq2d 6760	. . . . . . 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 7271	. . . . . 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 5170	. . . . 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 7271	. . . 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 6760	. . 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 1146	. . . . 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 21928	. . . . 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 21929	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) finSupp (0g‘𝐴))
30		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → ((coe1‘𝐾)‘𝑘) = ((coe1‘𝐾)‘𝑥))
31	30	oveq1d 7270	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (((coe1‘𝐾)‘𝑘) ∗ 𝑂) = (((coe1‘𝐾)‘𝑥) ∗ 𝑂))
32	31	cbvmptv 5183	. . . . . . . 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 5077	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) finSupp (0g‘𝐴) ↔ (𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂)) finSupp (0g‘𝐴))
35	33, 34	anbi12i 626	. . . . . 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 21882	. . . . . 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 593	. . . . 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 833	. . . 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 6757	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) = ((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛)
46	45	fveq2i 6759	. . . . . . . 8 ⊢ (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛)) = (𝑇‘((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛))
47	46	oveq2i 7266	. . . . . . 7 ⊢ ((𝑛 ↑ 𝑋) · (𝑇‘((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛))) = ((𝑛 ↑ 𝑋) · (𝑇‘((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛)))
48	47	mpteq2i 5175	. . . . . 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 7266	. . . . 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 6759	. . . 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 7265	. . . . . 6 ⊢ (((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍)) = (((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍))
52	51	mpteq2i 5175	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍))) = (𝑛 ∈ ℕ0 ↦ (((𝑥 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑥) ∗ 𝑂))‘𝑛) ∙ (𝑛𝐸𝑍)))
53	52	oveq2i 7266	. . . 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 2804	. . 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 481	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂))‘𝑛) = (((coe1‘𝐾)‘𝑛) ∗ 𝑂))
57	56	oveq1d 7270	. . . 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 5170	. . 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 7271	. 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 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   class class class wbr 5070   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  Fincfn 8691   finSupp cfsupp 9058  ℕ₀cn0 12163  Basecbs 16840   ·_𝑠 cvsca 16892  0_gc0g 17067   Σ_g cgsu 17068  ._gcmg 18615  mulGrpcmgp 19635  1_rcur 19652  CRingccrg 19699  algSccascl 20969  var₁cv1 21257  Poly₁cpl1 21258  coe₁cco1 21259   Mat cmat 21464   matToPolyMat cmat2pmat 21761   pMatToMatPoly cpm2mp 21849   CharPlyMat cchpmat 21883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-addf 10881  ax-mulf 10882

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-xor 1504  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-ofr 7512  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-tpos 8013  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-sup 9131  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-xnn0 12236  df-z 12250  df-dec 12367  df-uz 12512  df-rp 12660  df-fz 13169  df-fzo 13312  df-seq 13650  df-exp 13711  df-hash 13973  df-word 14146  df-lsw 14194  df-concat 14202  df-s1 14229  df-substr 14282  df-pfx 14312  df-splice 14391  df-reverse 14400  df-s2 14489  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-starv 16903  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-unif 16911  df-hom 16912  df-cco 16913  df-0g 17069  df-gsum 17070  df-prds 17075  df-pws 17077  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-submnd 18346  df-efmnd 18423  df-grp 18495  df-minusg 18496  df-sbg 18497  df-mulg 18616  df-subg 18667  df-ghm 18747  df-gim 18790  df-cntz 18838  df-oppg 18865  df-symg 18890  df-pmtr 18965  df-psgn 19014  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-srg 19657  df-ring 19700  df-cring 19701  df-oppr 19777  df-dvdsr 19798  df-unit 19799  df-invr 19829  df-dvr 19840  df-rnghom 19874  df-drng 19908  df-subrg 19937  df-lmod 20040  df-lss 20109  df-sra 20349  df-rgmod 20350  df-cnfld 20511  df-zring 20583  df-zrh 20617  df-dsmm 20849  df-frlm 20864  df-assa 20970  df-ascl 20972  df-psr 21022  df-mvr 21023  df-mpl 21024  df-opsr 21026  df-psr1 21261  df-vr1 21262  df-ply1 21263  df-coe1 21264  df-mamu 21443  df-mat 21465  df-mdet 21642  df-mat2pmat 21764  df-decpmat 21820  df-pm2mp 21850  df-chpmat 21884

This theorem is referenced by:  chcoeffeq  21943