![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pm2mpcl | Structured version Visualization version GIF version |
Description: The transformation of polynomial matrices into polynomials over matrices maps polynomial matrices to polynomials over matrices. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 5-Dec-2019.) |
Ref | Expression |
---|---|
pm2mpval.p | ⊢ 𝑃 = (Poly1‘𝑅) |
pm2mpval.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
pm2mpval.b | ⊢ 𝐵 = (Base‘𝐶) |
pm2mpval.m | ⊢ ∗ = ( ·𝑠 ‘𝑄) |
pm2mpval.e | ⊢ ↑ = (.g‘(mulGrp‘𝑄)) |
pm2mpval.x | ⊢ 𝑋 = (var1‘𝐴) |
pm2mpval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
pm2mpval.q | ⊢ 𝑄 = (Poly1‘𝐴) |
pm2mpval.t | ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) |
pm2mpcl.l | ⊢ 𝐿 = (Base‘𝑄) |
Ref | Expression |
---|---|
pm2mpcl | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2mpval.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | pm2mpval.c | . . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
3 | pm2mpval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
4 | pm2mpval.m | . . 3 ⊢ ∗ = ( ·𝑠 ‘𝑄) | |
5 | pm2mpval.e | . . 3 ⊢ ↑ = (.g‘(mulGrp‘𝑄)) | |
6 | pm2mpval.x | . . 3 ⊢ 𝑋 = (var1‘𝐴) | |
7 | pm2mpval.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
8 | pm2mpval.q | . . 3 ⊢ 𝑄 = (Poly1‘𝐴) | |
9 | pm2mpval.t | . . 3 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | pm2mpfval 21008 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))) |
11 | pm2mpcl.l | . . 3 ⊢ 𝐿 = (Base‘𝑄) | |
12 | eqid 2778 | . . 3 ⊢ (0g‘𝑄) = (0g‘𝑄) | |
13 | 7 | matring 20653 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
14 | 8 | ply1ring 20014 | . . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring) |
15 | ringcmn 18968 | . . . . 5 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ CMnd) | |
16 | 13, 14, 15 | 3syl 18 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd) |
17 | 16 | 3adant3 1123 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑄 ∈ CMnd) |
18 | nn0ex 11649 | . . . 4 ⊢ ℕ0 ∈ V | |
19 | 18 | a1i 11 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ℕ0 ∈ V) |
20 | 13 | 3adant3 1123 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring) |
21 | 20 | adantr 474 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ Ring) |
22 | simpl2 1201 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring) | |
23 | simpl3 1203 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ 𝐵) | |
24 | simpr 479 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
25 | eqid 2778 | . . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
26 | 1, 2, 3, 7, 25 | decpmatcl 20979 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) → (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴)) |
27 | 22, 23, 24, 26 | syl3anc 1439 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴)) |
28 | eqid 2778 | . . . . . 6 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄) | |
29 | 25, 8, 6, 4, 28, 5, 11 | ply1tmcl 20038 | . . . . 5 ⊢ ((𝐴 ∈ Ring ∧ (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴) ∧ 𝑘 ∈ ℕ0) → ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) ∈ 𝐿) |
30 | 21, 27, 24, 29 | syl3anc 1439 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) ∈ 𝐿) |
31 | 30 | fmpttd 6649 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))):ℕ0⟶𝐿) |
32 | 8 | ply1lmod 20018 | . . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod) |
33 | 20, 32 | syl 17 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑄 ∈ LMod) |
34 | eqidd 2779 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑄) = (Scalar‘𝑄)) | |
35 | 8, 6, 28, 5, 11 | ply1moncl 20037 | . . . . 5 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ 𝐿) |
36 | 20, 35 | sylan 575 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ 𝐿) |
37 | eqid 2778 | . . . 4 ⊢ (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄)) | |
38 | eqid 2778 | . . . . . . 7 ⊢ (0g‘𝐴) = (0g‘𝐴) | |
39 | 1, 2, 3, 7, 38 | decpmatfsupp 20981 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑀 decompPMat 𝑘)) finSupp (0g‘𝐴)) |
40 | 39 | 3adant1 1121 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑀 decompPMat 𝑘)) finSupp (0g‘𝐴)) |
41 | 8 | ply1sca 20019 | . . . . . . . 8 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄)) |
42 | 41 | eqcomd 2784 | . . . . . . 7 ⊢ (𝐴 ∈ Ring → (Scalar‘𝑄) = 𝐴) |
43 | 20, 42 | syl 17 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑄) = 𝐴) |
44 | 43 | fveq2d 6450 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (0g‘(Scalar‘𝑄)) = (0g‘𝐴)) |
45 | 40, 44 | breqtrrd 4914 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑀 decompPMat 𝑘)) finSupp (0g‘(Scalar‘𝑄))) |
46 | 19, 33, 34, 11, 27, 36, 12, 37, 4, 45 | mptscmfsupp0 19320 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) finSupp (0g‘𝑄)) |
47 | 11, 12, 17, 19, 31, 46 | gsumcl 18702 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) ∈ 𝐿) |
48 | 10, 47 | eqeltrd 2859 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 Vcvv 3398 class class class wbr 4886 ↦ cmpt 4965 ‘cfv 6135 (class class class)co 6922 Fincfn 8241 finSupp cfsupp 8563 ℕ0cn0 11642 Basecbs 16255 Scalarcsca 16341 ·𝑠 cvsca 16342 0gc0g 16486 Σg cgsu 16487 .gcmg 17927 CMndccmn 18579 mulGrpcmgp 18876 Ringcrg 18934 LModclmod 19255 var1cv1 19942 Poly1cpl1 19943 Mat cmat 20617 decompPMat cdecpmat 20974 pMatToMatPoly cpm2mp 21004 |
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 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
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 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-ofr 7175 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-hom 16362 df-cco 16363 df-0g 16488 df-gsum 16489 df-prds 16494 df-pws 16496 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-ghm 18042 df-cntz 18133 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-subrg 19170 df-lmod 19257 df-lss 19325 df-sra 19569 df-rgmod 19570 df-psr 19753 df-mvr 19754 df-mpl 19755 df-opsr 19757 df-psr1 19946 df-vr1 19947 df-ply1 19948 df-coe1 19949 df-dsmm 20475 df-frlm 20490 df-mamu 20594 df-mat 20618 df-decpmat 20975 df-pm2mp 21005 |
This theorem is referenced by: pm2mpf 21010 pm2mpf1 21011 |
Copyright terms: Public domain | W3C validator |