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Mirrors > Home > MPE Home > Th. List > pmatcollpw3fi | Structured version Visualization version GIF version |
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, 8-Dec-2019.) |
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) |
pmatcollpw3.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
pmatcollpw3.d | ⊢ 𝐷 = (Base‘𝐴) |
Ref | Expression |
---|---|
pmatcollpw3fi | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∃𝑓 ∈ (𝐷 ↑𝑚 (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmatcollpw.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | pmatcollpw.c | . . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
3 | pmatcollpw.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
4 | pmatcollpw.m | . . 3 ⊢ ∗ = ( ·𝑠 ‘𝐶) | |
5 | pmatcollpw.e | . . 3 ⊢ ↑ = (.g‘(mulGrp‘𝑃)) | |
6 | pmatcollpw.x | . . 3 ⊢ 𝑋 = (var1‘𝑅) | |
7 | pmatcollpw.t | . . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
8 | 1, 2, 3, 4, 5, 6, 7 | pmatcollpwfi 20998 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))))) |
9 | elnn0uz 12035 | . . . . . 6 ⊢ (𝑠 ∈ ℕ0 ↔ 𝑠 ∈ (ℤ≥‘0)) | |
10 | fzn0 12676 | . . . . . 6 ⊢ ((0...𝑠) ≠ ∅ ↔ 𝑠 ∈ (ℤ≥‘0)) | |
11 | 9, 10 | sylbb2 230 | . . . . 5 ⊢ (𝑠 ∈ ℕ0 → (0...𝑠) ≠ ∅) |
12 | fz0ssnn0 12757 | . . . . 5 ⊢ (0...𝑠) ⊆ ℕ0 | |
13 | 11, 12 | jctil 515 | . . . 4 ⊢ (𝑠 ∈ ℕ0 → ((0...𝑠) ⊆ ℕ0 ∧ (0...𝑠) ≠ ∅)) |
14 | pmatcollpw3.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
15 | pmatcollpw3.d | . . . . 5 ⊢ 𝐷 = (Base‘𝐴) | |
16 | 1, 2, 3, 4, 5, 6, 7, 14, 15 | pmatcollpw3lem 20999 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ ((0...𝑠) ⊆ ℕ0 ∧ (0...𝑠) ≠ ∅)) → (𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))) → ∃𝑓 ∈ (𝐷 ↑𝑚 (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))))) |
17 | 13, 16 | sylan2 586 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) → (𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))) → ∃𝑓 ∈ (𝐷 ↑𝑚 (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))))) |
18 | 17 | reximdva 3198 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ ℕ0 𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))) → ∃𝑠 ∈ ℕ0 ∃𝑓 ∈ (𝐷 ↑𝑚 (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))))) |
19 | 8, 18 | mpd 15 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∃𝑓 ∈ (𝐷 ↑𝑚 (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∃wrex 3091 ⊆ wss 3792 ∅c0 4141 ↦ cmpt 4967 ‘cfv 6137 (class class class)co 6924 ↑𝑚 cmap 8142 Fincfn 8243 0cc0 10274 ℕ0cn0 11646 ℤ≥cuz 11996 ...cfz 12647 Basecbs 16259 ·𝑠 cvsca 16346 Σg cgsu 16491 .gcmg 17931 mulGrpcmgp 18880 CRingccrg 18939 var1cv1 19946 Poly1cpl1 19947 Mat cmat 20621 matToPolyMat cmat2pmat 20920 decompPMat cdecpmat 20978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-ot 4407 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-ofr 7177 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-cur 7677 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-sup 8638 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-fz 12648 df-fzo 12789 df-seq 13124 df-hash 13440 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-sca 16358 df-vsca 16359 df-ip 16360 df-tset 16361 df-ple 16362 df-ds 16364 df-hom 16366 df-cco 16367 df-0g 16492 df-gsum 16493 df-prds 16498 df-pws 16500 df-mre 16636 df-mrc 16637 df-acs 16639 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-mhm 17725 df-submnd 17726 df-grp 17816 df-minusg 17817 df-sbg 17818 df-mulg 17932 df-subg 17979 df-ghm 18046 df-cntz 18137 df-cmn 18585 df-abl 18586 df-mgp 18881 df-ur 18893 df-srg 18897 df-ring 18940 df-cring 18941 df-subrg 19174 df-lmod 19261 df-lss 19329 df-sra 19573 df-rgmod 19574 df-assa 19713 df-ascl 19715 df-psr 19757 df-mvr 19758 df-mpl 19759 df-opsr 19761 df-psr1 19950 df-vr1 19951 df-ply1 19952 df-coe1 19953 df-dsmm 20479 df-frlm 20494 df-mamu 20598 df-mat 20622 df-mat2pmat 20923 df-decpmat 20979 |
This theorem is referenced by: pmatcollpw3fi1 21004 |
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