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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 = (Poly1𝑅)
pmatcollpw.c 𝐶 = (𝑁 Mat 𝑃)
pmatcollpw.b 𝐵 = (Base‘𝐶)
pmatcollpw.m = ( ·𝑠𝐶)
pmatcollpw.e = (.g‘(mulGrp‘𝑃))
pmatcollpw.x 𝑋 = (var1𝑅)
pmatcollpw.t 𝑇 = (𝑁 matToPolyMat 𝑅)
Assertion
Ref Expression
pmatcollpwfi ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃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
StepHypRef Expression
1 crngring 20197 . . . 4 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
213ad2ant2 1131 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
3 simp3 1135 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀𝐵)
4 pmatcollpw.p . . . 4 𝑃 = (Poly1𝑅)
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))
94, 5, 6, 7, 8decpmataa0 22714 . . 3 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))))
102, 3, 9syl2anc 582 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))))
11 pmatcollpw.m . . . . . . 7 = ( ·𝑠𝐶)
12 pmatcollpw.e . . . . . . 7 = (.g‘(mulGrp‘𝑃))
13 pmatcollpw.x . . . . . . 7 𝑋 = (var1𝑅)
14 pmatcollpw.t . . . . . . 7 𝑇 = (𝑁 matToPolyMat 𝑅)
154, 5, 6, 11, 12, 13, 14pmatcollpw 22727 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))))
1615ad2antrr 724 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 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 (0g𝐶) = (0g𝐶)
18 simp1 1133 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑁 ∈ Fin)
194, 5pmatring 22638 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
2018, 2, 19syl2anc 582 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐶 ∈ Ring)
21 ringcmn 20230 . . . . . . . 8 (𝐶 ∈ Ring → 𝐶 ∈ CMnd)
2220, 21syl 17 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐶 ∈ CMnd)
2322ad2antrr 724 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅)))) → 𝐶 ∈ CMnd)
2418adantr 479 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
252adantr 479 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
264ply1ring 22190 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
2725, 26syl 17 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑃 ∈ Ring)
282anim1i 613 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0))
29 eqid 2725 . . . . . . . . . . 11 (mulGrp‘𝑃) = (mulGrp‘𝑃)
30 eqid 2725 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘𝑃)
314, 13, 29, 12, 30ply1moncl 22215 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ (Base‘𝑃))
3228, 31syl 17 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ (Base‘𝑃))
33 simpl2 1189 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ CRing)
343adantr 479 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑀𝐵)
35 simpr 483 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
36 eqid 2725 . . . . . . . . . . . 12 (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
374, 5, 6, 7, 36decpmatcl 22713 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑀𝐵𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
3833, 34, 35, 37syl3anc 1368 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
3914, 7, 36, 4, 5, 6mat2pmatbas0 22673 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅))) → (𝑇‘(𝑀 decompPMat 𝑛)) ∈ 𝐵)
4024, 25, 38, 39syl3anc 1368 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑀 decompPMat 𝑛)) ∈ 𝐵)
4130, 5, 6, 11matvscl 22377 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝑛 𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘(𝑀 decompPMat 𝑛)) ∈ 𝐵)) → ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵)
4224, 27, 32, 40, 41syl22anc 837 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵)
4342ralrimiva 3135 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∀𝑛 ∈ ℕ0 ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵)
4443ad2antrr 724 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 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 ∧ 𝑀𝐵) ∧ 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))))
472, 18jca 510 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
4847ad2antrr 724 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
49 eqid 2725 . . . . . . . . . . . . . . 15 (0g‘(𝑁 Mat 𝑃)) = (0g‘(𝑁 Mat 𝑃))
5014, 4, 8, 490mat2pmat 22682 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘(0g‘(𝑁 Mat 𝑅))) = (0g‘(𝑁 Mat 𝑃)))
5148, 50syl 17 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(0g‘(𝑁 Mat 𝑅))) = (0g‘(𝑁 Mat 𝑃)))
5246, 51sylan9eqr 2787 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) ∧ (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → (𝑇‘(𝑀 decompPMat 𝑛)) = (0g‘(𝑁 Mat 𝑃)))
5352oveq2d 7435 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) ∧ (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))) = ((𝑛 𝑋) (0g‘(𝑁 Mat 𝑃))))
544, 5pmatlmod 22639 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
5518, 2, 54syl2anc 582 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐶 ∈ LMod)
5655ad2antrr 724 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝐶 ∈ LMod)
5728adantlr 713 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0))
5857, 31syl 17 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ (Base‘𝑃))
594ply1crng 22141 . . . . . . . . . . . . . . . . . . . 20 (𝑅 ∈ CRing → 𝑃 ∈ CRing)
6059anim2i 615 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
61603adant3 1129 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
625matsca2 22366 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝐶))
6361, 62syl 17 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑃 = (Scalar‘𝐶))
6463eqcomd 2731 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Scalar‘𝐶) = 𝑃)
6564ad2antrr 724 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (Scalar‘𝐶) = 𝑃)
6665fveq2d 6900 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
6758, 66eleqtrrd 2828 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)))
685eqcomi 2734 . . . . . . . . . . . . . . . 16 (𝑁 Mat 𝑃) = 𝐶
6968fveq2i 6899 . . . . . . . . . . . . . . 15 (0g‘(𝑁 Mat 𝑃)) = (0g𝐶)
7069oveq2i 7430 . . . . . . . . . . . . . 14 ((𝑛 𝑋) (0g‘(𝑁 Mat 𝑃))) = ((𝑛 𝑋) (0g𝐶))
71 eqid 2725 . . . . . . . . . . . . . . 15 (Scalar‘𝐶) = (Scalar‘𝐶)
72 eqid 2725 . . . . . . . . . . . . . . 15 (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
7371, 11, 72, 17lmodvs0 20791 . . . . . . . . . . . . . 14 ((𝐶 ∈ LMod ∧ (𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶))) → ((𝑛 𝑋) (0g𝐶)) = (0g𝐶))
7470, 73eqtrid 2777 . . . . . . . . . . . . 13 ((𝐶 ∈ LMod ∧ (𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶))) → ((𝑛 𝑋) (0g‘(𝑁 Mat 𝑃))) = (0g𝐶))
7556, 67, 74syl2anc 582 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑋) (0g‘(𝑁 Mat 𝑃))) = (0g𝐶))
7675adantr 479 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) ∧ (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → ((𝑛 𝑋) (0g‘(𝑁 Mat 𝑃))) = (0g𝐶))
7753, 76eqtrd 2765 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) ∧ (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))) = (0g𝐶))
7877ex 411 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅)) → ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))) = (0g𝐶)))
7978imim2d 57 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → (𝑠 < 𝑛 → ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))) = (0g𝐶))))
8079ralimdva 3156 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))) = (0g𝐶))))
8180imp 405 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅)))) → ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))) = (0g𝐶)))
826, 17, 23, 44, 45, 81gsummptnn0fz 19953 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 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))))))
8316, 82eqtrd 2765 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 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))))))
8483ex 411 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 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)))))))
8584reximdva 3157 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑀 decompPMat 𝑛) = (0g‘(𝑁 Mat 𝑅))) → ∃𝑠 ∈ ℕ0 𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))))))
8610, 85mpd 15 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ0 𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))))
Colors of variables: wff setvar class
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  0cn0 12505  ...cfz 13519  Basecbs 17183  Scalarcsca 17239   ·𝑠 cvsca 17240  0gc0g 17424   Σg 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
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