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Theorem pmatcollpw 21389
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 = (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
pmatcollpw ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 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.
StepHypRef Expression
1 crngring 19309 . . 3 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2 pmatcollpw.p . . . 4 𝑃 = (Poly1𝑅)
3 pmatcollpw.c . . . 4 𝐶 = (𝑁 Mat 𝑃)
4 pmatcollpw.b . . . 4 𝐵 = (Base‘𝐶)
5 eqid 2824 . . . 4 ( ·𝑠𝑃) = ( ·𝑠𝑃)
6 pmatcollpw.e . . . 4 = (.g‘(mulGrp‘𝑃))
7 pmatcollpw.x . . . 4 𝑋 = (var1𝑅)
82, 3, 4, 5, 6, 7pmatcollpw2 21386 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋))))))
91, 8syl3an2 1161 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋))))))
10 eqidd 2825 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎𝑁𝑏𝑁)) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋))))
11 oveq12 7158 . . . . . . . . . 10 ((𝑖 = 𝑎𝑗 = 𝑏) → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑎(𝑀 decompPMat 𝑛)𝑏))
1211oveq1d 7164 . . . . . . . . 9 ((𝑖 = 𝑎𝑗 = 𝑏) → ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋)) = ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠𝑃)(𝑛 𝑋)))
1312adantl 485 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎𝑁𝑏𝑁)) ∧ (𝑖 = 𝑎𝑗 = 𝑏)) → ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋)) = ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠𝑃)(𝑛 𝑋)))
14 simprl 770 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎𝑁𝑏𝑁)) → 𝑎𝑁)
15 simpr 488 . . . . . . . . 9 ((𝑎𝑁𝑏𝑁) → 𝑏𝑁)
1615adantl 485 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎𝑁𝑏𝑁)) → 𝑏𝑁)
17 simp2 1134 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ CRing)
1817adantr 484 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ CRing)
1918, 1syl 17 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
2019adantr 484 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎𝑁𝑏𝑁)) → 𝑅 ∈ Ring)
21 eqid 2824 . . . . . . . . . 10 (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
22 eqid 2824 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
23 eqid 2824 . . . . . . . . . 10 (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
24 simp3 1135 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀𝐵)
2524adantr 484 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑀𝐵)
26 simpr 488 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
272, 3, 4, 21, 23decpmatcl 21375 . . . . . . . . . . . 12 ((𝑅 ∈ CRing ∧ 𝑀𝐵𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
2818, 25, 26, 27syl3anc 1368 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
2928adantr 484 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎𝑁𝑏𝑁)) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
3021, 22, 23, 14, 16, 29matecld 21035 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅))
31 simplr 768 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎𝑁𝑏𝑁)) → 𝑛 ∈ ℕ0)
32 eqid 2824 . . . . . . . . . 10 (mulGrp‘𝑃) = (mulGrp‘𝑃)
33 eqid 2824 . . . . . . . . . 10 (Base‘𝑃) = (Base‘𝑃)
3422, 2, 7, 5, 32, 6, 33ply1tmcl 20440 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅) ∧ 𝑛 ∈ ℕ0) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠𝑃)(𝑛 𝑋)) ∈ (Base‘𝑃))
3520, 30, 31, 34syl3anc 1368 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎𝑁𝑏𝑁)) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠𝑃)(𝑛 𝑋)) ∈ (Base‘𝑃))
3610, 13, 14, 16, 35ovmpod 7295 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋)))𝑏) = ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠𝑃)(𝑛 𝑋)))
37 pmatcollpw.m . . . . . . . . 9 = ( ·𝑠𝐶)
38 pmatcollpw.t . . . . . . . . 9 𝑇 = (𝑁 matToPolyMat 𝑅)
392, 3, 4, 37, 6, 7, 38pmatcollpwlem 21388 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠𝑃)(𝑛 𝑋)) = (𝑎((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
40393expb 1117 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎𝑁𝑏𝑁)) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠𝑃)(𝑛 𝑋)) = (𝑎((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
4136, 40eqtrd 2859 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋)))𝑏) = (𝑎((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
4241ralrimivva 3186 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → ∀𝑎𝑁𝑏𝑁 (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋)))𝑏) = (𝑎((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
43 simpl1 1188 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
442ply1ring 20416 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
451, 44syl 17 . . . . . . . . 9 (𝑅 ∈ CRing → 𝑃 ∈ Ring)
46453ad2ant2 1131 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑃 ∈ Ring)
4746adantr 484 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑃 ∈ Ring)
48193ad2ant1 1130 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
49 simp2 1134 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
50 simp3 1135 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
51283ad2ant1 1130 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
5221, 22, 23, 49, 50, 51matecld 21035 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (𝑖(𝑀 decompPMat 𝑛)𝑗) ∈ (Base‘𝑅))
53263ad2ant1 1130 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑛 ∈ ℕ0)
5422, 2, 7, 5, 32, 6, 33ply1tmcl 20440 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑖(𝑀 decompPMat 𝑛)𝑗) ∈ (Base‘𝑅) ∧ 𝑛 ∈ ℕ0) → ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋)) ∈ (Base‘𝑃))
5548, 52, 53, 54syl3anc 1368 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋)) ∈ (Base‘𝑃))
563, 33, 4, 43, 47, 55matbas2d 21032 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋))) ∈ 𝐵)
5713ad2ant2 1131 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
582, 7, 32, 6, 33ply1moncl 20439 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ (Base‘𝑃))
5957, 58sylan 583 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ (Base‘𝑃))
6057adantr 484 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
6138, 21, 23, 2, 3mat2pmatbas 21334 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅))) → (𝑇‘(𝑀 decompPMat 𝑛)) ∈ (Base‘𝐶))
6243, 60, 28, 61syl3anc 1368 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑀 decompPMat 𝑛)) ∈ (Base‘𝐶))
6362, 4eleqtrrdi 2927 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑀 decompPMat 𝑛)) ∈ 𝐵)
6433, 3, 4, 37matvscl 21040 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝑛 𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘(𝑀 decompPMat 𝑛)) ∈ 𝐵)) → ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵)
6543, 47, 59, 63, 64syl22anc 837 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵)
663, 4eqmat 21033 . . . . . 6 (((𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋))) ∈ 𝐵 ∧ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))) ∈ 𝐵) → ((𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋))) = ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))) ↔ ∀𝑎𝑁𝑏𝑁 (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋)))𝑏) = (𝑎((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))𝑏)))
6756, 65, 66syl2anc 587 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋))) = ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))) ↔ ∀𝑎𝑁𝑏𝑁 (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋)))𝑏) = (𝑎((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))𝑏)))
6842, 67mpbird 260 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋))) = ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))
6968mpteq2dva 5147 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋)))) = (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))))
7069oveq2d 7165 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐶 Σg (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗)( ·𝑠𝑃)(𝑛 𝑋))))) = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))))
719, 70eqtrd 2859 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  cmpt 5132  cfv 6343  (class class class)co 7149  cmpo 7151  Fincfn 8505  0cn0 11894  Basecbs 16483   ·𝑠 cvsca 16569   Σg cgsu 16714  .gcmg 18224  mulGrpcmgp 19239  Ringcrg 19297  CRingccrg 19298  var1cv1 20344  Poly1cpl1 20345   Mat cmat 21016   matToPolyMat cmat2pmat 21312   decompPMat cdecpmat 21370
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-ot 4559  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-ofr 7404  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fsupp 8831  df-sup 8903  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-dec 12096  df-uz 12241  df-fz 12895  df-fzo 13038  df-seq 13374  df-hash 13696  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-hom 16589  df-cco 16590  df-0g 16715  df-gsum 16716  df-prds 16721  df-pws 16723  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-submnd 17957  df-grp 18106  df-minusg 18107  df-sbg 18108  df-mulg 18225  df-subg 18276  df-ghm 18356  df-cntz 18447  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-srg 19256  df-ring 19299  df-cring 19300  df-subrg 19533  df-lmod 19636  df-lss 19704  df-sra 19944  df-rgmod 19945  df-assa 20085  df-ascl 20087  df-psr 20136  df-mvr 20137  df-mpl 20138  df-opsr 20140  df-psr1 20348  df-vr1 20349  df-ply1 20350  df-coe1 20351  df-dsmm 20876  df-frlm 20891  df-mat 21017  df-mat2pmat 21315  df-decpmat 21371
This theorem is referenced by:  pmatcollpwfi  21390  pmatcollpw3  21392  pmatcollpwscmat  21399
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