![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > chpdmatlem0 | Structured version Visualization version GIF version |
Description: Lemma 0 for chpdmat 21023. (Contributed by AV, 18-Aug-2019.) |
Ref | Expression |
---|---|
chpdmat.c | ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) |
chpdmat.p | ⊢ 𝑃 = (Poly1‘𝑅) |
chpdmat.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
chpdmat.s | ⊢ 𝑆 = (algSc‘𝑃) |
chpdmat.b | ⊢ 𝐵 = (Base‘𝐴) |
chpdmat.x | ⊢ 𝑋 = (var1‘𝑅) |
chpdmat.0 | ⊢ 0 = (0g‘𝑅) |
chpdmat.g | ⊢ 𝐺 = (mulGrp‘𝑃) |
chpdmat.m | ⊢ − = (-g‘𝑃) |
chpdmatlem.q | ⊢ 𝑄 = (𝑁 Mat 𝑃) |
chpdmatlem.1 | ⊢ 1 = (1r‘𝑄) |
chpdmatlem.m | ⊢ · = ( ·𝑠 ‘𝑄) |
Ref | Expression |
---|---|
chpdmatlem0 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 · 1 ) ∈ (Base‘𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chpdmat.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | chpdmatlem.q | . . 3 ⊢ 𝑄 = (𝑁 Mat 𝑃) | |
3 | 1, 2 | pmatlmod 20876 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod) |
4 | chpdmat.x | . . . . 5 ⊢ 𝑋 = (var1‘𝑅) | |
5 | eqid 2825 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
6 | 4, 1, 5 | vr1cl 19954 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
7 | 6 | adantl 475 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑋 ∈ (Base‘𝑃)) |
8 | 1 | ply1ring 19985 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
9 | 2 | matsca2 20600 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑃 = (Scalar‘𝑄)) |
10 | 8, 9 | sylan2 586 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 = (Scalar‘𝑄)) |
11 | 10 | eqcomd 2831 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝑄) = 𝑃) |
12 | 11 | fveq2d 6441 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑄)) = (Base‘𝑃)) |
13 | 7, 12 | eleqtrrd 2909 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑋 ∈ (Base‘(Scalar‘𝑄))) |
14 | 1, 2 | pmatring 20875 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring) |
15 | eqid 2825 | . . . 4 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
16 | chpdmatlem.1 | . . . 4 ⊢ 1 = (1r‘𝑄) | |
17 | 15, 16 | ringidcl 18929 | . . 3 ⊢ (𝑄 ∈ Ring → 1 ∈ (Base‘𝑄)) |
18 | 14, 17 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 ∈ (Base‘𝑄)) |
19 | eqid 2825 | . . 3 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄) | |
20 | chpdmatlem.m | . . 3 ⊢ · = ( ·𝑠 ‘𝑄) | |
21 | eqid 2825 | . . 3 ⊢ (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄)) | |
22 | 15, 19, 20, 21 | lmodvscl 19243 | . 2 ⊢ ((𝑄 ∈ LMod ∧ 𝑋 ∈ (Base‘(Scalar‘𝑄)) ∧ 1 ∈ (Base‘𝑄)) → (𝑋 · 1 ) ∈ (Base‘𝑄)) |
23 | 3, 13, 18, 22 | syl3anc 1494 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 · 1 ) ∈ (Base‘𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 Fincfn 8228 Basecbs 16229 Scalarcsca 16315 ·𝑠 cvsca 16316 0gc0g 16460 -gcsg 17785 mulGrpcmgp 18850 1rcur 18862 Ringcrg 18908 LModclmod 19226 algSccascl 19679 var1cv1 19913 Poly1cpl1 19914 Mat cmat 20587 CharPlyMat cchpmat 21008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-ot 4408 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-ofr 7163 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-sup 8623 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-fzo 12768 df-seq 13103 df-hash 13418 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-hom 16336 df-cco 16337 df-0g 16462 df-gsum 16463 df-prds 16468 df-pws 16470 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-mulg 17902 df-subg 17949 df-ghm 18016 df-cntz 18107 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-subrg 19141 df-lmod 19228 df-lss 19296 df-sra 19540 df-rgmod 19541 df-psr 19724 df-mvr 19725 df-mpl 19726 df-opsr 19728 df-psr1 19917 df-vr1 19918 df-ply1 19919 df-dsmm 20446 df-frlm 20461 df-mamu 20564 df-mat 20588 |
This theorem is referenced by: chpdmatlem1 21020 chpdmatlem2 21021 chpdmatlem3 21022 |
Copyright terms: Public domain | W3C validator |