Theorem pmatcollpwfi 21912

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 19776	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	3ad2ant2 1132	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
3		simp3 1136	. . 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 2739	. . . 4 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
8		eqid 2739	. . . 4 ⊢ (0g‘(𝑁 Mat 𝑅)) = (0g‘(𝑁 Mat 𝑅))
9	4, 5, 6, 7, 8	decpmataa0 21898	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))))
10	2, 3, 9	syl2anc 583	. 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 21911	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))
16	15	ad2antrr 722	. . . . 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 2739	. . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶)
18		simp1 1134	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
19	4, 5	pmatring 21822	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
20	18, 2, 19	syl2anc 583	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐶 ∈ Ring)
21		ringcmn 19801	. . . . . . . 8 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ CMnd)
22	20, 21	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐶 ∈ CMnd)
23	22	ad2antrr 722	. . . . . 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 480	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
25	2	adantr 480	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
26	4	ply1ring 21400	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
27	25, 26	syl 17	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑃 ∈ Ring)
28	2	anim1i 614	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0))
29		eqid 2739	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
30		eqid 2739	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
31	4, 13, 29, 12, 30	ply1moncl 21423	. . . . . . . . . 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 1190	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ CRing)
34	3	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ 𝐵)
35		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
36		eqid 2739	. . . . . . . . . . . 12 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
37	4, 5, 6, 7, 36	decpmatcl 21897	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
38	33, 34, 35, 37	syl3anc 1369	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
39	14, 7, 36, 4, 5, 6	mat2pmatbas0 21857	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅))) → (𝑇‘(𝑀 decompPMat 𝑛)) ∈ 𝐵)
40	24, 25, 38, 39	syl3anc 1369	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑀 decompPMat 𝑛)) ∈ 𝐵)
41	30, 5, 6, 11	matvscl 21561	. . . . . . . . 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 835	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵)
43	42	ralrimiva 3109	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ ℕ0 ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵)
44	43	ad2antrr 722	. . . . . 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 765	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅)))) → 𝑠 ∈ ℕ0)
46		fveq2 6768	. . . . . . . . . . . . 13 ⊢ ((𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅)) → (𝑇‘(𝑀 decompPMat 𝑛)) = (𝑇‘(0g‘(𝑁 Mat 𝑅))))
47	2, 18	jca 511	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
48	47	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
49		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (0g‘(𝑁 Mat 𝑃)) = (0g‘(𝑁 Mat 𝑃))
50	14, 4, 8, 49	0mat2pmat 21866	. . . . . . . . . . . . . 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 2801	. . . . . . . . . . . 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 7284	. . . . . . . . . . 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 21823	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
55	18, 2, 54	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐶 ∈ LMod)
56	55	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝐶 ∈ LMod)
57	28	adantlr 711	. . . . . . . . . . . . . . 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 21350	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
60	59	anim2i 616	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
61	60	3adant3 1130	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
62	5	matsca2 21550	. . . . . . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝐶) = 𝑃)
65	64	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (Scalar‘𝐶) = 𝑃)
66	65	fveq2d 6772	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
67	58, 66	eleqtrrd 2843	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶)))
68	5	eqcomi 2748	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 Mat 𝑃) = 𝐶
69	68	fveq2i 6771	. . . . . . . . . . . . . . 15 ⊢ (0g‘(𝑁 Mat 𝑃)) = (0g‘𝐶)
70	69	oveq2i 7279	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ↑ 𝑋) ∗ (0g‘(𝑁 Mat 𝑃))) = ((𝑛 ↑ 𝑋) ∗ (0g‘𝐶))
71		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
72		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
73	71, 11, 72, 17	lmodvs0 20138	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶))) → ((𝑛 ↑ 𝑋) ∗ (0g‘𝐶)) = (0g‘𝐶))
74	70, 73	eqtrid 2791	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶))) → ((𝑛 ↑ 𝑋) ∗ (0g‘(𝑁 Mat 𝑃))) = (0g‘𝐶))
75	56, 67, 74	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑋) ∗ (0g‘(𝑁 Mat 𝑃))) = (0g‘𝐶))
76	75	adantr 480	. . . . . . . . . . 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 2779	. . . . . . . . . 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 412	. . . . . . . . 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 3104	. . . . . . 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 406	. . . . . 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 19568	. . . . 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 2779	. . . 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 412	. . 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 3204	. 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 395   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  ∀wral 3065  ∃wrex 3066   class class class wbr 5078   ↦ cmpt 5161  ‘cfv 6430  (class class class)co 7268  Fincfn 8707  0cc0 10855   < clt 10993  ℕ₀cn0 12216  ...cfz 13221  Basecbs 16893  Scalarcsca 16946   ·_𝑠 cvsca 16947  0_gc0g 17131   Σ_g cgsu 17132  ._gcmg 18681  CMndccmn 19367  mulGrpcmgp 19701  Ringcrg 19764  CRingccrg 19765  LModclmod 20104  var₁cv1 21328  Poly₁cpl1 21329   Mat cmat 21535   matToPolyMat cmat2pmat 21834   decompPMat cdecpmat 21892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-ot 4575  df-uni 4845  df-int 4885  df-iun 4931  df-iin 4932  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-se 5544  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-isom 6439  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-of 7524  df-ofr 7525  df-om 7701  df-1st 7817  df-2nd 7818  df-supp 7962  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-1o 8281  df-er 8472  df-map 8591  df-pm 8592  df-ixp 8660  df-en 8708  df-dom 8709  df-sdom 8710  df-fin 8711  df-fsupp 9090  df-sup 9162  df-oi 9230  df-card 9681  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-nn 11957  df-2 12019  df-3 12020  df-4 12021  df-5 12022  df-6 12023  df-7 12024  df-8 12025  df-9 12026  df-n0 12217  df-z 12303  df-dec 12420  df-uz 12565  df-fz 13222  df-fzo 13365  df-seq 13703  df-hash 14026  df-struct 16829  df-sets 16846  df-slot 16864  df-ndx 16876  df-base 16894  df-ress 16923  df-plusg 16956  df-mulr 16957  df-sca 16959  df-vsca 16960  df-ip 16961  df-tset 16962  df-ple 16963  df-ds 16965  df-hom 16967  df-cco 16968  df-0g 17133  df-gsum 17134  df-prds 17139  df-pws 17141  df-mre 17276  df-mrc 17277  df-acs 17279  df-mgm 18307  df-sgrp 18356  df-mnd 18367  df-mhm 18411  df-submnd 18412  df-grp 18561  df-minusg 18562  df-sbg 18563  df-mulg 18682  df-subg 18733  df-ghm 18813  df-cntz 18904  df-cmn 19369  df-abl 19370  df-mgp 19702  df-ur 19719  df-srg 19723  df-ring 19766  df-cring 19767  df-subrg 20003  df-lmod 20106  df-lss 20175  df-sra 20415  df-rgmod 20416  df-dsmm 20920  df-frlm 20935  df-assa 21041  df-ascl 21043  df-psr 21093  df-mvr 21094  df-mpl 21095  df-opsr 21097  df-psr1 21332  df-vr1 21333  df-ply1 21334  df-coe1 21335  df-mamu 21514  df-mat 21536  df-mat2pmat 21837  df-decpmat 21893

This theorem is referenced by:  pmatcollpw3fi  21915