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Mirrors > Home > MPE Home > Th. List > pmatcollpwscmat | Structured version Visualization version GIF version |
Description: Write a scalar matrix over polynomials (over a commutative ring) as a sum of the product of variable powers and constant scalar matrices with scalar entries. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.) |
Ref | Expression |
---|---|
pmatcollpwscmat.p | ⊢ 𝑃 = (Poly1‘𝑅) |
pmatcollpwscmat.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
pmatcollpwscmat.b | ⊢ 𝐵 = (Base‘𝐶) |
pmatcollpwscmat.m1 | ⊢ ∗ = ( ·𝑠 ‘𝐶) |
pmatcollpwscmat.e1 | ⊢ ↑ = (.g‘(mulGrp‘𝑃)) |
pmatcollpwscmat.x | ⊢ 𝑋 = (var1‘𝑅) |
pmatcollpwscmat.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
pmatcollpwscmat.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
pmatcollpwscmat.d | ⊢ 𝐷 = (Base‘𝐴) |
pmatcollpwscmat.u | ⊢ 𝑈 = (algSc‘𝑃) |
pmatcollpwscmat.k | ⊢ 𝐾 = (Base‘𝑅) |
pmatcollpwscmat.e2 | ⊢ 𝐸 = (Base‘𝑃) |
pmatcollpwscmat.s | ⊢ 𝑆 = (algSc‘𝑃) |
pmatcollpwscmat.1 | ⊢ 1 = (1r‘𝐶) |
pmatcollpwscmat.m2 | ⊢ 𝑀 = (𝑄 ∗ 1 ) |
Ref | Expression |
---|---|
pmatcollpwscmat | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄 ∈ 𝐸) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ ((𝑈‘((coe1‘𝑄)‘𝑛)) ∗ 1 ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 19021 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | pmatcollpwscmat.m2 | . . . . 5 ⊢ 𝑀 = (𝑄 ∗ 1 ) | |
3 | pmatcollpwscmat.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | pmatcollpwscmat.c | . . . . . 6 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
5 | pmatcollpwscmat.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
6 | pmatcollpwscmat.e2 | . . . . . 6 ⊢ 𝐸 = (Base‘𝑃) | |
7 | pmatcollpwscmat.m1 | . . . . . 6 ⊢ ∗ = ( ·𝑠 ‘𝐶) | |
8 | pmatcollpwscmat.1 | . . . . . 6 ⊢ 1 = (1r‘𝐶) | |
9 | 3, 4, 5, 6, 7, 8 | 1pmatscmul 21004 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) → (𝑄 ∗ 1 ) ∈ 𝐵) |
10 | 2, 9 | syl5eqel 2864 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) → 𝑀 ∈ 𝐵) |
11 | 1, 10 | syl3an2 1144 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄 ∈ 𝐸) → 𝑀 ∈ 𝐵) |
12 | pmatcollpwscmat.e1 | . . . 4 ⊢ ↑ = (.g‘(mulGrp‘𝑃)) | |
13 | pmatcollpwscmat.x | . . . 4 ⊢ 𝑋 = (var1‘𝑅) | |
14 | pmatcollpwscmat.t | . . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
15 | 3, 4, 5, 7, 12, 13, 14 | pmatcollpw 21083 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))))) |
16 | 11, 15 | syld3an3 1389 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄 ∈ 𝐸) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))))) |
17 | 1 | anim2i 607 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
18 | 17 | 3adant3 1112 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄 ∈ 𝐸) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
19 | simp3 1118 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄 ∈ 𝐸) → 𝑄 ∈ 𝐸) | |
20 | 19 | anim1ci 606 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄 ∈ 𝐸) ∧ 𝑛 ∈ ℕ0) → (𝑛 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) |
21 | pmatcollpwscmat.a | . . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
22 | pmatcollpwscmat.d | . . . . . . 7 ⊢ 𝐷 = (Base‘𝐴) | |
23 | pmatcollpwscmat.u | . . . . . . 7 ⊢ 𝑈 = (algSc‘𝑃) | |
24 | pmatcollpwscmat.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝑅) | |
25 | pmatcollpwscmat.s | . . . . . . 7 ⊢ 𝑆 = (algSc‘𝑃) | |
26 | 3, 4, 5, 7, 12, 13, 14, 21, 22, 23, 24, 6, 25, 8, 2 | pmatcollpwscmatlem2 21092 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑛 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑇‘(𝑀 decompPMat 𝑛)) = ((𝑈‘((coe1‘𝑄)‘𝑛)) ∗ 1 )) |
27 | 18, 20, 26 | syl2an2r 672 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄 ∈ 𝐸) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑀 decompPMat 𝑛)) = ((𝑈‘((coe1‘𝑄)‘𝑛)) ∗ 1 )) |
28 | 27 | oveq2d 6986 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄 ∈ 𝐸) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))) = ((𝑛 ↑ 𝑋) ∗ ((𝑈‘((coe1‘𝑄)‘𝑛)) ∗ 1 ))) |
29 | 28 | mpteq2dva 5016 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄 ∈ 𝐸) → (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))) = (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ ((𝑈‘((coe1‘𝑄)‘𝑛)) ∗ 1 )))) |
30 | 29 | oveq2d 6986 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄 ∈ 𝐸) → (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))) = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ ((𝑈‘((coe1‘𝑄)‘𝑛)) ∗ 1 ))))) |
31 | 16, 30 | eqtrd 2808 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄 ∈ 𝐸) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ ((𝑈‘((coe1‘𝑄)‘𝑛)) ∗ 1 ))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 ↦ cmpt 5002 ‘cfv 6182 (class class class)co 6970 Fincfn 8298 ℕ0cn0 11700 Basecbs 16329 ·𝑠 cvsca 16415 Σg cgsu 16560 .gcmg 18001 mulGrpcmgp 18952 1rcur 18964 Ringcrg 19010 CRingccrg 19011 algSccascl 19795 var1cv1 20037 Poly1cpl1 20038 coe1cco1 20039 Mat cmat 20710 matToPolyMat cmat2pmat 21006 decompPMat cdecpmat 21064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-ot 4444 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-ofr 7222 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-pm 8201 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-sup 8693 df-oi 8761 df-card 9154 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-fz 12702 df-fzo 12843 df-seq 13178 df-hash 13499 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-hom 16435 df-cco 16436 df-0g 16561 df-gsum 16562 df-prds 16567 df-pws 16569 df-mre 16705 df-mrc 16706 df-acs 16708 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-mhm 17793 df-submnd 17794 df-grp 17884 df-minusg 17885 df-sbg 17886 df-mulg 18002 df-subg 18050 df-ghm 18117 df-cntz 18208 df-cmn 18658 df-abl 18659 df-mgp 18953 df-ur 18965 df-srg 18969 df-ring 19012 df-cring 19013 df-subrg 19246 df-lmod 19348 df-lss 19416 df-sra 19656 df-rgmod 19657 df-assa 19796 df-ascl 19798 df-psr 19840 df-mvr 19841 df-mpl 19842 df-opsr 19844 df-psr1 20041 df-vr1 20042 df-ply1 20043 df-coe1 20044 df-dsmm 20568 df-frlm 20583 df-mamu 20687 df-mat 20711 df-mat2pmat 21009 df-decpmat 21065 |
This theorem is referenced by: cpmidgsum 21170 |
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