Theorem pmatcollpw 22274

Description: Write a polynomial matrix (over a commutative ring) as a sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 26-Oct-2019.) (Revised by AV, 4-Dec-2019.)

Hypotheses

Ref	Expression
pmatcollpw.p	⊢ 𝑃 = (Poly1‘𝑅)
pmatcollpw.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pmatcollpw.b	⊢ 𝐵 = (Base‘𝐶)
pmatcollpw.m	⊢ ∗ = ( ·𝑠 ‘𝐶)
pmatcollpw.e	⊢ ↑ = (.g‘(mulGrp‘𝑃))
pmatcollpw.x	⊢ 𝑋 = (var1‘𝑅)
pmatcollpw.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)

Assertion

Ref	Expression
pmatcollpw	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))

Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑃,𝑛   𝑅,𝑛   𝑛,𝑋   ↑ ,𝑛

Allowed substitution hints:   𝐶(𝑛)   𝑇(𝑛)   ∗ (𝑛)

Proof of Theorem pmatcollpw

Dummy variables 𝑖 𝑗 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngring 20061	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		pmatcollpw.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
3		pmatcollpw.c	. . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃)
4		pmatcollpw.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
5		eqid 2732	. . . 4 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
6		pmatcollpw.e	. . . 4 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
7		pmatcollpw.x	. . . 4 ⊢ 𝑋 = (var1‘𝑅)
8	2, 3, 4, 5, 6, 7	pmatcollpw2 22271	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋))))))
9	1, 8	syl3an2 1164	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋))))))
10		eqidd 2733	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋))))
11		oveq12 7414	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑎(𝑀 decompPMat 𝑛)𝑏))
12	11	oveq1d 7420	. . . . . . . . 9 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) = ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)))
13	12	adantl 482	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) ∧ (𝑖 = 𝑎 ∧ 𝑗 = 𝑏)) → ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) = ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)))
14		simprl 769	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑎 ∈ 𝑁)
15		simpr 485	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑏 ∈ 𝑁)
16	15	adantl 482	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑏 ∈ 𝑁)
17		simp2 1137	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ CRing)
18	17	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ CRing)
19	18, 1	syl 17	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
20	19	adantr 481	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑅 ∈ Ring)
21		eqid 2732	. . . . . . . . . 10 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
22		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
23		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
24		simp3 1138	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
25	24	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ 𝐵)
26		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
27	2, 3, 4, 21, 23	decpmatcl 22260	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
28	18, 25, 26, 27	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
29	28	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
30	21, 22, 23, 14, 16, 29	matecld 21919	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅))
31		simplr 767	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → 𝑛 ∈ ℕ0)
32		eqid 2732	. . . . . . . . . 10 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
33		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘𝑃) = (Base‘𝑃)
34	22, 2, 7, 5, 32, 6, 33	ply1tmcl 21785	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅) ∧ 𝑛 ∈ ℕ0) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) ∈ (Base‘𝑃))
35	20, 30, 31, 34	syl3anc 1371	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) ∈ (Base‘𝑃))
36	10, 13, 14, 16, 35	ovmpod 7556	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)))𝑏) = ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)))
37		pmatcollpw.m	. . . . . . . . 9 ⊢ ∗ = ( ·𝑠 ‘𝐶)
38		pmatcollpw.t	. . . . . . . . 9 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
39	2, 3, 4, 37, 6, 7, 38	pmatcollpwlem 22273	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
40	39	3expb 1120	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
41	36, 40	eqtrd 2772	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)))𝑏) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
42	41	ralrimivva 3200	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)))𝑏) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
43		simpl1 1191	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
44	2	ply1ring 21761	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
45	1, 44	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
46	45	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
47	46	adantr 481	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑃 ∈ Ring)
48	19	3ad2ant1 1133	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
49		simp2 1137	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
50		simp3 1138	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
51	28	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
52	21, 22, 23, 49, 50, 51	matecld 21919	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑀 decompPMat 𝑛)𝑗) ∈ (Base‘𝑅))
53	26	3ad2ant1 1133	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑛 ∈ ℕ0)
54	22, 2, 7, 5, 32, 6, 33	ply1tmcl 21785	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑖(𝑀 decompPMat 𝑛)𝑗) ∈ (Base‘𝑅) ∧ 𝑛 ∈ ℕ0) → ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) ∈ (Base‘𝑃))
55	48, 52, 53, 54	syl3anc 1371	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) ∈ (Base‘𝑃))
56	3, 33, 4, 43, 47, 55	matbas2d 21916	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋))) ∈ 𝐵)
57	1	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
58	2, 7, 32, 6, 33	ply1moncl 21784	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
59	57, 58	sylan 580	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
60	57	adantr 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
61	38, 21, 23, 2, 3	mat2pmatbas 22219	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅))) → (𝑇‘(𝑀 decompPMat 𝑛)) ∈ (Base‘𝐶))
62	43, 60, 28, 61	syl3anc 1371	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑀 decompPMat 𝑛)) ∈ (Base‘𝐶))
63	62, 4	eleqtrrdi 2844	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑀 decompPMat 𝑛)) ∈ 𝐵)
64	33, 3, 4, 37	matvscl 21924	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝑛 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘(𝑀 decompPMat 𝑛)) ∈ 𝐵)) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵)
65	43, 47, 59, 63, 64	syl22anc 837	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵)
66	3, 4	eqmat 21917	. . . . . 6 ⊢ (((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋))) ∈ 𝐵 ∧ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋))) = ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) ↔ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)))𝑏) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏)))
67	56, 65, 66	syl2anc 584	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋))) = ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) ↔ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)))𝑏) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏)))
68	42, 67	mpbird 256	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋))) = ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))
69	68	mpteq2dva 5247	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)))) = (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))))
70	69	oveq2d 7421	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶 Σg (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋))))) = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))
71	9, 70	eqtrd 2772	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ↦ cmpt 5230  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407  Fincfn 8935  ℕ₀cn0 12468  Basecbs 17140   ·_𝑠 cvsca 17197   Σ_g cgsu 17382  ._gcmg 18944  mulGrpcmgp 19981  Ringcrg 20049  CRingccrg 20050  var₁cv1 21691  Poly₁cpl1 21692   Mat cmat 21898   matToPolyMat cmat2pmat 22197   decompPMat cdecpmat 22255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-ot 4636  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-ofr 7667  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-hom 17217  df-cco 17218  df-0g 17383  df-gsum 17384  df-prds 17389  df-pws 17391  df-mre 17526  df-mrc 17527  df-acs 17529  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-submnd 18668  df-grp 18818  df-minusg 18819  df-sbg 18820  df-mulg 18945  df-subg 18997  df-ghm 19084  df-cntz 19175  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-srg 20003  df-ring 20051  df-cring 20052  df-subrg 20353  df-lmod 20465  df-lss 20535  df-sra 20777  df-rgmod 20778  df-dsmm 21278  df-frlm 21293  df-assa 21399  df-ascl 21401  df-psr 21453  df-mvr 21454  df-mpl 21455  df-opsr 21457  df-psr1 21695  df-vr1 21696  df-ply1 21697  df-coe1 21698  df-mat 21899  df-mat2pmat 22200  df-decpmat 22256

This theorem is referenced by:  pmatcollpwfi  22275  pmatcollpw3  22277  pmatcollpwscmat  22284