![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mat2pmatscmxcl | Structured version Visualization version GIF version |
Description: A transformed matrix multiplied with a power of the variable of a polynomial is a polynomial matrix. (Contributed by AV, 6-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.) |
Ref | Expression |
---|---|
mat2pmatscmxcl.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mat2pmatscmxcl.k | ⊢ 𝐾 = (Base‘𝐴) |
mat2pmatscmxcl.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
mat2pmatscmxcl.p | ⊢ 𝑃 = (Poly1‘𝑅) |
mat2pmatscmxcl.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
mat2pmatscmxcl.b | ⊢ 𝐵 = (Base‘𝐶) |
mat2pmatscmxcl.m | ⊢ ∗ = ( ·𝑠 ‘𝐶) |
mat2pmatscmxcl.e | ⊢ ↑ = (.g‘(mulGrp‘𝑃)) |
mat2pmatscmxcl.x | ⊢ 𝑋 = (var1‘𝑅) |
Ref | Expression |
---|---|
mat2pmatscmxcl | ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → ((𝐿 ↑ 𝑋) ∗ (𝑇‘𝑀)) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 765 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → 𝑁 ∈ Fin) | |
2 | mat2pmatscmxcl.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | 2 | ply1ring 21615 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
4 | 3 | ad2antlr 725 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → 𝑃 ∈ Ring) |
5 | mat2pmatscmxcl.x | . . . 4 ⊢ 𝑋 = (var1‘𝑅) | |
6 | eqid 2736 | . . . 4 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃) | |
7 | mat2pmatscmxcl.e | . . . 4 ⊢ ↑ = (.g‘(mulGrp‘𝑃)) | |
8 | eqid 2736 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
9 | 2, 5, 6, 7, 8 | ply1moncl 21638 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐿 ∈ ℕ0) → (𝐿 ↑ 𝑋) ∈ (Base‘𝑃)) |
10 | 9 | ad2ant2l 744 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝐿 ↑ 𝑋) ∈ (Base‘𝑃)) |
11 | simpl 483 | . . . . 5 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0) → 𝑀 ∈ 𝐾) | |
12 | 11 | anim2i 617 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ 𝐾)) |
13 | df-3an 1089 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ 𝐾)) | |
14 | 12, 13 | sylibr 233 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾)) |
15 | mat2pmatscmxcl.t | . . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
16 | mat2pmatscmxcl.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
17 | mat2pmatscmxcl.k | . . . 4 ⊢ 𝐾 = (Base‘𝐴) | |
18 | mat2pmatscmxcl.c | . . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
19 | mat2pmatscmxcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
20 | 15, 16, 17, 2, 18, 19 | mat2pmatbas0 22072 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑇‘𝑀) ∈ 𝐵) |
21 | 14, 20 | syl 17 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝑇‘𝑀) ∈ 𝐵) |
22 | mat2pmatscmxcl.m | . . 3 ⊢ ∗ = ( ·𝑠 ‘𝐶) | |
23 | 8, 18, 19, 22 | matvscl 21776 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝐿 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑀) ∈ 𝐵)) → ((𝐿 ↑ 𝑋) ∗ (𝑇‘𝑀)) ∈ 𝐵) |
24 | 1, 4, 10, 21, 23 | syl22anc 837 | 1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → ((𝐿 ↑ 𝑋) ∗ (𝑇‘𝑀)) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6494 (class class class)co 7354 Fincfn 8880 ℕ0cn0 12410 Basecbs 17080 ·𝑠 cvsca 17134 .gcmg 18868 mulGrpcmgp 19892 Ringcrg 19960 var1cv1 21543 Poly1cpl1 21544 Mat cmat 21750 matToPolyMat cmat2pmat 22049 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7614 df-ofr 7615 df-om 7800 df-1st 7918 df-2nd 7919 df-supp 8090 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-er 8645 df-map 8764 df-pm 8765 df-ixp 8833 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-fsupp 9303 df-sup 9375 df-oi 9443 df-card 9872 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12411 df-z 12497 df-dec 12616 df-uz 12761 df-fz 13422 df-fzo 13565 df-seq 13904 df-hash 14228 df-struct 17016 df-sets 17033 df-slot 17051 df-ndx 17063 df-base 17081 df-ress 17110 df-plusg 17143 df-mulr 17144 df-sca 17146 df-vsca 17147 df-ip 17148 df-tset 17149 df-ple 17150 df-ds 17152 df-hom 17154 df-cco 17155 df-0g 17320 df-gsum 17321 df-prds 17326 df-pws 17328 df-mre 17463 df-mrc 17464 df-acs 17466 df-mgm 18494 df-sgrp 18543 df-mnd 18554 df-mhm 18598 df-submnd 18599 df-grp 18748 df-minusg 18749 df-sbg 18750 df-mulg 18869 df-subg 18921 df-ghm 19002 df-cntz 19093 df-cmn 19560 df-abl 19561 df-mgp 19893 df-ur 19910 df-ring 19962 df-subrg 20216 df-lmod 20320 df-lss 20389 df-sra 20629 df-rgmod 20630 df-dsmm 21134 df-frlm 21149 df-ascl 21257 df-psr 21307 df-mvr 21308 df-mpl 21309 df-opsr 21311 df-psr1 21547 df-vr1 21548 df-ply1 21549 df-mat 21751 df-mat2pmat 22052 |
This theorem is referenced by: pmatcollpw3fi1lem1 22131 monmat2matmon 22169 pm2mp 22170 cpmadugsumlemF 22221 cpmadugsumfi 22222 |
Copyright terms: Public domain | W3C validator |