Theorem pmatcollpwfi 22728

Description: Write a polynomial matrix (over a commutative ring) as a finite sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 4-Nov-2019.) (Revised by AV, 4-Dec-2019.) (Proof shortened by AV, 3-Jul-2022.)

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
pmatcollpwfi	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))

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

Allowed substitution hints:   𝐶(𝑠)   𝑃(𝑠)   𝑇(𝑛,𝑠)   ↑ (𝑠)   ∗ (𝑛,𝑠)   𝑋(𝑠)

Proof of Theorem pmatcollpwfi

Step	Hyp	Ref	Expression
1		crngring 20197	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	3ad2ant2 1131	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
3		simp3 1135	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
4		pmatcollpw.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
5		pmatcollpw.c	. . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃)
6		pmatcollpw.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
7		eqid 2725	. . . 4 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
8		eqid 2725	. . . 4 ⊢ (0g‘(𝑁 Mat 𝑅)) = (0g‘(𝑁 Mat 𝑅))
9	4, 5, 6, 7, 8	decpmataa0 22714	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))))
10	2, 3, 9	syl2anc 582	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))))
11		pmatcollpw.m	. . . . . . 7 ⊢ ∗ = ( ·𝑠 ‘𝐶)
12		pmatcollpw.e	. . . . . . 7 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
13		pmatcollpw.x	. . . . . . 7 ⊢ 𝑋 = (var1‘𝑅)
14		pmatcollpw.t	. . . . . . 7 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
15	4, 5, 6, 11, 12, 13, 14	pmatcollpw 22727	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))
16	15	ad2antrr 724	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅)))) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))
17		eqid 2725	. . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶)
18		simp1 1133	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
19	4, 5	pmatring 22638	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
20	18, 2, 19	syl2anc 582	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐶 ∈ Ring)
21		ringcmn 20230	. . . . . . . 8 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ CMnd)
22	20, 21	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐶 ∈ CMnd)
23	22	ad2antrr 724	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅)))) → 𝐶 ∈ CMnd)
24	18	adantr 479	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
25	2	adantr 479	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
26	4	ply1ring 22190	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
27	25, 26	syl 17	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑃 ∈ Ring)
28	2	anim1i 613	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0))
29		eqid 2725	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
30		eqid 2725	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
31	4, 13, 29, 12, 30	ply1moncl 22215	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
32	28, 31	syl 17	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
33		simpl2 1189	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ CRing)
34	3	adantr 479	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ 𝐵)
35		simpr 483	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
36		eqid 2725	. . . . . . . . . . . 12 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
37	4, 5, 6, 7, 36	decpmatcl 22713	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
38	33, 34, 35, 37	syl3anc 1368	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
39	14, 7, 36, 4, 5, 6	mat2pmatbas0 22673	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅))) → (𝑇‘(𝑀 decompPMat 𝑛)) ∈ 𝐵)
40	24, 25, 38, 39	syl3anc 1368	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑀 decompPMat 𝑛)) ∈ 𝐵)
41	30, 5, 6, 11	matvscl 22377	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝑛 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘(𝑀 decompPMat 𝑛)) ∈ 𝐵)) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵)
42	24, 27, 32, 40, 41	syl22anc 837	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵)
43	42	ralrimiva 3135	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ ℕ0 ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵)
44	43	ad2antrr 724	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅)))) → ∀𝑛 ∈ ℕ0 ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵)
45		simplr 767	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅)))) → 𝑠 ∈ ℕ0)
46		fveq2 6896	. . . . . . . . . . . . 13 ⊢ ((𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅)) → (𝑇‘(𝑀 decompPMat 𝑛)) = (𝑇‘(0g‘(𝑁 Mat 𝑅))))
47	2, 18	jca 510	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
48	47	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
49		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (0g‘(𝑁 Mat 𝑃)) = (0g‘(𝑁 Mat 𝑃))
50	14, 4, 8, 49	0mat2pmat 22682	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘(0g‘(𝑁 Mat 𝑅))) = (0g‘(𝑁 Mat 𝑃)))
51	48, 50	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(0g‘(𝑁 Mat 𝑅))) = (0g‘(𝑁 Mat 𝑃)))
52	46, 51	sylan9eqr 2787	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) ∧ (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → (𝑇‘(𝑀 decompPMat 𝑛)) = (0g‘(𝑁 Mat 𝑃)))
53	52	oveq2d 7435	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) ∧ (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) = ((𝑛 ↑ 𝑋) ∗ (0g‘(𝑁 Mat 𝑃))))
54	4, 5	pmatlmod 22639	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
55	18, 2, 54	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐶 ∈ LMod)
56	55	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝐶 ∈ LMod)
57	28	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0))
58	57, 31	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
59	4	ply1crng 22141	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
60	59	anim2i 615	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
61	60	3adant3 1129	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
62	5	matsca2 22366	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝐶))
63	61, 62	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 = (Scalar‘𝐶))
64	63	eqcomd 2731	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝐶) = 𝑃)
65	64	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (Scalar‘𝐶) = 𝑃)
66	65	fveq2d 6900	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
67	58, 66	eleqtrrd 2828	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶)))
68	5	eqcomi 2734	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 Mat 𝑃) = 𝐶
69	68	fveq2i 6899	. . . . . . . . . . . . . . 15 ⊢ (0g‘(𝑁 Mat 𝑃)) = (0g‘𝐶)
70	69	oveq2i 7430	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ↑ 𝑋) ∗ (0g‘(𝑁 Mat 𝑃))) = ((𝑛 ↑ 𝑋) ∗ (0g‘𝐶))
71		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
72		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
73	71, 11, 72, 17	lmodvs0 20791	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶))) → ((𝑛 ↑ 𝑋) ∗ (0g‘𝐶)) = (0g‘𝐶))
74	70, 73	eqtrid 2777	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶))) → ((𝑛 ↑ 𝑋) ∗ (0g‘(𝑁 Mat 𝑃))) = (0g‘𝐶))
75	56, 67, 74	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑋) ∗ (0g‘(𝑁 Mat 𝑃))) = (0g‘𝐶))
76	75	adantr 479	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) ∧ (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → ((𝑛 ↑ 𝑋) ∗ (0g‘(𝑁 Mat 𝑃))) = (0g‘𝐶))
77	53, 76	eqtrd 2765	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) ∧ (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) = (0g‘𝐶))
78	77	ex 411	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅)) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) = (0g‘𝐶)))
79	78	imim2d 57	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → (𝑠 < 𝑛 → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) = (0g‘𝐶))))
80	79	ralimdva 3156	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) = (0g‘𝐶))))
81	80	imp 405	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅)))) → ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) = (0g‘𝐶)))
82	6, 17, 23, 44, 45, 81	gsummptnn0fz 19953	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅)))) → (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))
83	16, 82	eqtrd 2765	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅)))) → 𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))
84	83	ex 411	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → 𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))))))
85	84	reximdva 3157	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → ∃𝑠 ∈ ℕ0 𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))))))
86	10, 85	mpd 15	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3050  ∃wrex 3059   class class class wbr 5149   ↦ cmpt 5232  ‘cfv 6549  (class class class)co 7419  Fincfn 8964  0cc0 11140   < clt 11280  ℕ₀cn0 12505  ...cfz 13519  Basecbs 17183  Scalarcsca 17239   ·_𝑠 cvsca 17240  0_gc0g 17424   Σ_g cgsu 17425  ._gcmg 19031  CMndccmn 19747  mulGrpcmgp 20086  Ringcrg 20185  CRingccrg 20186  LModclmod 20755  var₁cv1 22118  Poly₁cpl1 22119   Mat cmat 22351   matToPolyMat cmat2pmat 22650   decompPMat cdecpmat 22708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-ot 4639  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-ofr 7686  df-om 7872  df-1st 7994  df-2nd 7995  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9388  df-sup 9467  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-uz 12856  df-fz 13520  df-fzo 13663  df-seq 14003  df-hash 14326  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-sca 17252  df-vsca 17253  df-ip 17254  df-tset 17255  df-ple 17256  df-ds 17258  df-hom 17260  df-cco 17261  df-0g 17426  df-gsum 17427  df-prds 17432  df-pws 17434  df-mre 17569  df-mrc 17570  df-acs 17572  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mhm 18743  df-submnd 18744  df-grp 18901  df-minusg 18902  df-sbg 18903  df-mulg 19032  df-subg 19086  df-ghm 19176  df-cntz 19280  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-srg 20139  df-ring 20187  df-cring 20188  df-subrng 20495  df-subrg 20520  df-lmod 20757  df-lss 20828  df-sra 21070  df-rgmod 21071  df-dsmm 21683  df-frlm 21698  df-assa 21804  df-ascl 21806  df-psr 21859  df-mvr 21860  df-mpl 21861  df-opsr 21863  df-psr1 22122  df-vr1 22123  df-ply1 22124  df-coe1 22125  df-mamu 22335  df-mat 22352  df-mat2pmat 22653  df-decpmat 22709

This theorem is referenced by:  pmatcollpw3fi  22731