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Theorem pmatcollpwfi 22504
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 = (Poly1β€˜π‘…)
pmatcollpw.c 𝐢 = (𝑁 Mat 𝑃)
pmatcollpw.b 𝐡 = (Baseβ€˜πΆ)
pmatcollpw.m βˆ— = ( ·𝑠 β€˜πΆ)
pmatcollpw.e ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))
pmatcollpw.x 𝑋 = (var1β€˜π‘…)
pmatcollpw.t 𝑇 = (𝑁 matToPolyMat 𝑅)
Assertion
Ref Expression
pmatcollpwfi ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„•0 𝑀 = (𝐢 Ξ£g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))))))
Distinct variable groups:   𝐡,𝑛   𝑛,𝑀   𝑛,𝑁   𝑃,𝑛   𝑅,𝑛   𝑛,𝑋   ↑ ,𝑛   𝐡,𝑠,𝑛   𝐢,𝑛   𝑀,𝑠   𝑁,𝑠   𝑅,𝑠
Allowed substitution hints:   𝐢(𝑠)   𝑃(𝑠)   𝑇(𝑛,𝑠)   ↑ (𝑠)   βˆ— (𝑛,𝑠)   𝑋(𝑠)

Proof of Theorem pmatcollpwfi
StepHypRef Expression
1 crngring 20139 . . . 4 (𝑅 ∈ CRing β†’ 𝑅 ∈ Ring)
213ad2ant2 1132 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ 𝑅 ∈ Ring)
3 simp3 1136 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ 𝑀 ∈ 𝐡)
4 pmatcollpw.p . . . 4 𝑃 = (Poly1β€˜π‘…)
5 pmatcollpw.c . . . 4 𝐢 = (𝑁 Mat 𝑃)
6 pmatcollpw.b . . . 4 𝐡 = (Baseβ€˜πΆ)
7 eqid 2730 . . . 4 (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
8 eqid 2730 . . . 4 (0gβ€˜(𝑁 Mat 𝑅)) = (0gβ€˜(𝑁 Mat 𝑅))
94, 5, 6, 7, 8decpmataa0 22490 . . 3 ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅))))
102, 3, 9syl2anc 582 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅))))
11 pmatcollpw.m . . . . . . 7 βˆ— = ( ·𝑠 β€˜πΆ)
12 pmatcollpw.e . . . . . . 7 ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))
13 pmatcollpw.x . . . . . . 7 𝑋 = (var1β€˜π‘…)
14 pmatcollpw.t . . . . . . 7 𝑇 = (𝑁 matToPolyMat 𝑅)
154, 5, 6, 11, 12, 13, 14pmatcollpw 22503 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ 𝑀 = (𝐢 Ξ£g (𝑛 ∈ β„•0 ↦ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))))))
1615ad2antrr 722 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅)))) β†’ 𝑀 = (𝐢 Ξ£g (𝑛 ∈ β„•0 ↦ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))))))
17 eqid 2730 . . . . . 6 (0gβ€˜πΆ) = (0gβ€˜πΆ)
18 simp1 1134 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ 𝑁 ∈ Fin)
194, 5pmatring 22414 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝐢 ∈ Ring)
2018, 2, 19syl2anc 582 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ 𝐢 ∈ Ring)
21 ringcmn 20170 . . . . . . . 8 (𝐢 ∈ Ring β†’ 𝐢 ∈ CMnd)
2220, 21syl 17 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ 𝐢 ∈ CMnd)
2322ad2antrr 722 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅)))) β†’ 𝐢 ∈ CMnd)
2418adantr 479 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑛 ∈ β„•0) β†’ 𝑁 ∈ Fin)
252adantr 479 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑛 ∈ β„•0) β†’ 𝑅 ∈ Ring)
264ply1ring 21990 . . . . . . . . . 10 (𝑅 ∈ Ring β†’ 𝑃 ∈ Ring)
2725, 26syl 17 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑛 ∈ β„•0) β†’ 𝑃 ∈ Ring)
282anim1i 613 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑛 ∈ β„•0) β†’ (𝑅 ∈ Ring ∧ 𝑛 ∈ β„•0))
29 eqid 2730 . . . . . . . . . . 11 (mulGrpβ€˜π‘ƒ) = (mulGrpβ€˜π‘ƒ)
30 eqid 2730 . . . . . . . . . . 11 (Baseβ€˜π‘ƒ) = (Baseβ€˜π‘ƒ)
314, 13, 29, 12, 30ply1moncl 22013 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑛 ∈ β„•0) β†’ (𝑛 ↑ 𝑋) ∈ (Baseβ€˜π‘ƒ))
3228, 31syl 17 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑛 ∈ β„•0) β†’ (𝑛 ↑ 𝑋) ∈ (Baseβ€˜π‘ƒ))
33 simpl2 1190 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑛 ∈ β„•0) β†’ 𝑅 ∈ CRing)
343adantr 479 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑛 ∈ β„•0) β†’ 𝑀 ∈ 𝐡)
35 simpr 483 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ β„•0)
36 eqid 2730 . . . . . . . . . . . 12 (Baseβ€˜(𝑁 Mat 𝑅)) = (Baseβ€˜(𝑁 Mat 𝑅))
374, 5, 6, 7, 36decpmatcl 22489 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡 ∧ 𝑛 ∈ β„•0) β†’ (𝑀 decompPMat 𝑛) ∈ (Baseβ€˜(𝑁 Mat 𝑅)))
3833, 34, 35, 37syl3anc 1369 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑛 ∈ β„•0) β†’ (𝑀 decompPMat 𝑛) ∈ (Baseβ€˜(𝑁 Mat 𝑅)))
3914, 7, 36, 4, 5, 6mat2pmatbas0 22449 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝑛) ∈ (Baseβ€˜(𝑁 Mat 𝑅))) β†’ (π‘‡β€˜(𝑀 decompPMat 𝑛)) ∈ 𝐡)
4024, 25, 38, 39syl3anc 1369 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑛 ∈ β„•0) β†’ (π‘‡β€˜(𝑀 decompPMat 𝑛)) ∈ 𝐡)
4130, 5, 6, 11matvscl 22153 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝑛 ↑ 𝑋) ∈ (Baseβ€˜π‘ƒ) ∧ (π‘‡β€˜(𝑀 decompPMat 𝑛)) ∈ 𝐡)) β†’ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))) ∈ 𝐡)
4224, 27, 32, 40, 41syl22anc 835 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))) ∈ 𝐡)
4342ralrimiva 3144 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ βˆ€π‘› ∈ β„•0 ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))) ∈ 𝐡)
4443ad2antrr 722 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅)))) β†’ βˆ€π‘› ∈ β„•0 ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))) ∈ 𝐡)
45 simplr 765 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅)))) β†’ 𝑠 ∈ β„•0)
46 fveq2 6890 . . . . . . . . . . . . 13 ((𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅)) β†’ (π‘‡β€˜(𝑀 decompPMat 𝑛)) = (π‘‡β€˜(0gβ€˜(𝑁 Mat 𝑅))))
472, 18jca 510 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
4847ad2antrr 722 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
49 eqid 2730 . . . . . . . . . . . . . . 15 (0gβ€˜(𝑁 Mat 𝑃)) = (0gβ€˜(𝑁 Mat 𝑃))
5014, 4, 8, 490mat2pmat 22458 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) β†’ (π‘‡β€˜(0gβ€˜(𝑁 Mat 𝑅))) = (0gβ€˜(𝑁 Mat 𝑃)))
5148, 50syl 17 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (π‘‡β€˜(0gβ€˜(𝑁 Mat 𝑅))) = (0gβ€˜(𝑁 Mat 𝑃)))
5246, 51sylan9eqr 2792 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) ∧ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅))) β†’ (π‘‡β€˜(𝑀 decompPMat 𝑛)) = (0gβ€˜(𝑁 Mat 𝑃)))
5352oveq2d 7427 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) ∧ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅))) β†’ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))) = ((𝑛 ↑ 𝑋) βˆ— (0gβ€˜(𝑁 Mat 𝑃))))
544, 5pmatlmod 22415 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝐢 ∈ LMod)
5518, 2, 54syl2anc 582 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ 𝐢 ∈ LMod)
5655ad2antrr 722 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ 𝐢 ∈ LMod)
5728adantlr 711 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝑅 ∈ Ring ∧ 𝑛 ∈ β„•0))
5857, 31syl 17 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝑛 ↑ 𝑋) ∈ (Baseβ€˜π‘ƒ))
594ply1crng 21941 . . . . . . . . . . . . . . . . . . . 20 (𝑅 ∈ CRing β†’ 𝑃 ∈ CRing)
6059anim2i 615 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) β†’ (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
61603adant3 1130 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
625matsca2 22142 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) β†’ 𝑃 = (Scalarβ€˜πΆ))
6361, 62syl 17 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ 𝑃 = (Scalarβ€˜πΆ))
6463eqcomd 2736 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ (Scalarβ€˜πΆ) = 𝑃)
6564ad2antrr 722 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (Scalarβ€˜πΆ) = 𝑃)
6665fveq2d 6894 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (Baseβ€˜(Scalarβ€˜πΆ)) = (Baseβ€˜π‘ƒ))
6758, 66eleqtrrd 2834 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝑛 ↑ 𝑋) ∈ (Baseβ€˜(Scalarβ€˜πΆ)))
685eqcomi 2739 . . . . . . . . . . . . . . . 16 (𝑁 Mat 𝑃) = 𝐢
6968fveq2i 6893 . . . . . . . . . . . . . . 15 (0gβ€˜(𝑁 Mat 𝑃)) = (0gβ€˜πΆ)
7069oveq2i 7422 . . . . . . . . . . . . . 14 ((𝑛 ↑ 𝑋) βˆ— (0gβ€˜(𝑁 Mat 𝑃))) = ((𝑛 ↑ 𝑋) βˆ— (0gβ€˜πΆ))
71 eqid 2730 . . . . . . . . . . . . . . 15 (Scalarβ€˜πΆ) = (Scalarβ€˜πΆ)
72 eqid 2730 . . . . . . . . . . . . . . 15 (Baseβ€˜(Scalarβ€˜πΆ)) = (Baseβ€˜(Scalarβ€˜πΆ))
7371, 11, 72, 17lmodvs0 20650 . . . . . . . . . . . . . 14 ((𝐢 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ (Baseβ€˜(Scalarβ€˜πΆ))) β†’ ((𝑛 ↑ 𝑋) βˆ— (0gβ€˜πΆ)) = (0gβ€˜πΆ))
7470, 73eqtrid 2782 . . . . . . . . . . . . 13 ((𝐢 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ (Baseβ€˜(Scalarβ€˜πΆ))) β†’ ((𝑛 ↑ 𝑋) βˆ— (0gβ€˜(𝑁 Mat 𝑃))) = (0gβ€˜πΆ))
7556, 67, 74syl2anc 582 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 ↑ 𝑋) βˆ— (0gβ€˜(𝑁 Mat 𝑃))) = (0gβ€˜πΆ))
7675adantr 479 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) ∧ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅))) β†’ ((𝑛 ↑ 𝑋) βˆ— (0gβ€˜(𝑁 Mat 𝑃))) = (0gβ€˜πΆ))
7753, 76eqtrd 2770 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) ∧ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅))) β†’ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))) = (0gβ€˜πΆ))
7877ex 411 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ ((𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅)) β†’ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))) = (0gβ€˜πΆ)))
7978imim2d 57 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ 𝑛 ∈ β„•0) β†’ ((𝑠 < 𝑛 β†’ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅))) β†’ (𝑠 < 𝑛 β†’ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))) = (0gβ€˜πΆ))))
8079ralimdva 3165 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) β†’ (βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅))) β†’ βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))) = (0gβ€˜πΆ))))
8180imp 405 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅)))) β†’ βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))) = (0gβ€˜πΆ)))
826, 17, 23, 44, 45, 81gsummptnn0fz 19895 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅)))) β†’ (𝐢 Ξ£g (𝑛 ∈ β„•0 ↦ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))))) = (𝐢 Ξ£g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))))))
8316, 82eqtrd 2770 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) ∧ βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅)))) β†’ 𝑀 = (𝐢 Ξ£g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))))))
8483ex 411 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•0) β†’ (βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅))) β†’ 𝑀 = (𝐢 Ξ£g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛)))))))
8584reximdva 3166 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ (βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (𝑀 decompPMat 𝑛) = (0gβ€˜(𝑁 Mat 𝑅))) β†’ βˆƒπ‘  ∈ β„•0 𝑀 = (𝐢 Ξ£g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛)))))))
8610, 85mpd 15 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„•0 𝑀 = (𝐢 Ξ£g (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) βˆ— (π‘‡β€˜(𝑀 decompPMat 𝑛))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068   class class class wbr 5147   ↦ cmpt 5230  β€˜cfv 6542  (class class class)co 7411  Fincfn 8941  0cc0 11112   < clt 11252  β„•0cn0 12476  ...cfz 13488  Basecbs 17148  Scalarcsca 17204   ·𝑠 cvsca 17205  0gc0g 17389   Ξ£g cgsu 17390  .gcmg 18986  CMndccmn 19689  mulGrpcmgp 20028  Ringcrg 20127  CRingccrg 20128  LModclmod 20614  var1cv1 21919  Poly1cpl1 21920   Mat cmat 22127   matToPolyMat cmat2pmat 22426   decompPMat cdecpmat 22484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-ofr 7673  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-sup 9439  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13489  df-fzo 13632  df-seq 13971  df-hash 14295  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ds 17223  df-hom 17225  df-cco 17226  df-0g 17391  df-gsum 17392  df-prds 17397  df-pws 17399  df-mre 17534  df-mrc 17535  df-acs 17537  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18705  df-submnd 18706  df-grp 18858  df-minusg 18859  df-sbg 18860  df-mulg 18987  df-subg 19039  df-ghm 19128  df-cntz 19222  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-srg 20081  df-ring 20129  df-cring 20130  df-subrng 20434  df-subrg 20459  df-lmod 20616  df-lss 20687  df-sra 20930  df-rgmod 20931  df-dsmm 21506  df-frlm 21521  df-assa 21627  df-ascl 21629  df-psr 21681  df-mvr 21682  df-mpl 21683  df-opsr 21685  df-psr1 21923  df-vr1 21924  df-ply1 21925  df-coe1 21926  df-mamu 22106  df-mat 22128  df-mat2pmat 22429  df-decpmat 22485
This theorem is referenced by:  pmatcollpw3fi  22507
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