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Mirrors > Home > MPE Home > Th. List > chpdmatlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for chpdmat 20978. (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) |
chpdmatlem.z | ⊢ 𝑍 = (-g‘𝑄) |
chpdmatlem.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
Ref | Expression |
---|---|
chpdmatlem1 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑋 · 1 )𝑍(𝑇‘𝑀)) ∈ (Base‘𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chpdmat.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | chpdmatlem.q | . . . . 5 ⊢ 𝑄 = (𝑁 Mat 𝑃) | |
3 | 1, 2 | pmatring 20830 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring) |
4 | 3 | 3adant3 1163 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑄 ∈ Ring) |
5 | ringgrp 18872 | . . 3 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ Grp) | |
6 | 4, 5 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑄 ∈ Grp) |
7 | chpdmat.c | . . . 4 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) | |
8 | chpdmat.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
9 | chpdmat.s | . . . 4 ⊢ 𝑆 = (algSc‘𝑃) | |
10 | chpdmat.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
11 | chpdmat.x | . . . 4 ⊢ 𝑋 = (var1‘𝑅) | |
12 | chpdmat.0 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
13 | chpdmat.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑃) | |
14 | chpdmat.m | . . . 4 ⊢ − = (-g‘𝑃) | |
15 | chpdmatlem.1 | . . . 4 ⊢ 1 = (1r‘𝑄) | |
16 | chpdmatlem.m | . . . 4 ⊢ · = ( ·𝑠 ‘𝑄) | |
17 | 7, 1, 8, 9, 10, 11, 12, 13, 14, 2, 15, 16 | chpdmatlem0 20974 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 · 1 ) ∈ (Base‘𝑄)) |
18 | 17 | 3adant3 1163 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑋 · 1 ) ∈ (Base‘𝑄)) |
19 | chpdmatlem.t | . . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
20 | 19, 8, 10, 1, 2 | mat2pmatbas 20863 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑄)) |
21 | eqid 2803 | . . 3 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
22 | chpdmatlem.z | . . 3 ⊢ 𝑍 = (-g‘𝑄) | |
23 | 21, 22 | grpsubcl 17815 | . 2 ⊢ ((𝑄 ∈ Grp ∧ (𝑋 · 1 ) ∈ (Base‘𝑄) ∧ (𝑇‘𝑀) ∈ (Base‘𝑄)) → ((𝑋 · 1 )𝑍(𝑇‘𝑀)) ∈ (Base‘𝑄)) |
24 | 6, 18, 20, 23 | syl3anc 1491 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑋 · 1 )𝑍(𝑇‘𝑀)) ∈ (Base‘𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ‘cfv 6105 (class class class)co 6882 Fincfn 8199 Basecbs 16188 ·𝑠 cvsca 16275 0gc0g 16419 Grpcgrp 17742 -gcsg 17744 mulGrpcmgp 18809 1rcur 18821 Ringcrg 18867 algSccascl 19638 var1cv1 19872 Poly1cpl1 19873 Mat cmat 20542 matToPolyMat cmat2pmat 20841 CharPlyMat cchpmat 20963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2379 ax-ext 2781 ax-rep 4968 ax-sep 4979 ax-nul 4987 ax-pow 5039 ax-pr 5101 ax-un 7187 ax-inf2 8792 ax-cnex 10284 ax-resscn 10285 ax-1cn 10286 ax-icn 10287 ax-addcl 10288 ax-addrcl 10289 ax-mulcl 10290 ax-mulrcl 10291 ax-mulcom 10292 ax-addass 10293 ax-mulass 10294 ax-distr 10295 ax-i2m1 10296 ax-1ne0 10297 ax-1rid 10298 ax-rnegex 10299 ax-rrecex 10300 ax-cnre 10301 ax-pre-lttri 10302 ax-pre-lttrn 10303 ax-pre-ltadd 10304 ax-pre-mulgt0 10305 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2593 df-eu 2611 df-clab 2790 df-cleq 2796 df-clel 2799 df-nfc 2934 df-ne 2976 df-nel 3079 df-ral 3098 df-rex 3099 df-reu 3100 df-rmo 3101 df-rab 3102 df-v 3391 df-sbc 3638 df-csb 3733 df-dif 3776 df-un 3778 df-in 3780 df-ss 3787 df-pss 3789 df-nul 4120 df-if 4282 df-pw 4355 df-sn 4373 df-pr 4375 df-tp 4377 df-op 4379 df-ot 4381 df-uni 4633 df-int 4672 df-iun 4716 df-iin 4717 df-br 4848 df-opab 4910 df-mpt 4927 df-tr 4950 df-id 5224 df-eprel 5229 df-po 5237 df-so 5238 df-fr 5275 df-se 5276 df-we 5277 df-xp 5322 df-rel 5323 df-cnv 5324 df-co 5325 df-dm 5326 df-rn 5327 df-res 5328 df-ima 5329 df-pred 5902 df-ord 5948 df-on 5949 df-lim 5950 df-suc 5951 df-iota 6068 df-fun 6107 df-fn 6108 df-f 6109 df-f1 6110 df-fo 6111 df-f1o 6112 df-fv 6113 df-isom 6114 df-riota 6843 df-ov 6885 df-oprab 6886 df-mpt2 6887 df-of 7135 df-ofr 7136 df-om 7304 df-1st 7405 df-2nd 7406 df-supp 7537 df-wrecs 7649 df-recs 7711 df-rdg 7749 df-1o 7803 df-2o 7804 df-oadd 7807 df-er 7986 df-map 8101 df-pm 8102 df-ixp 8153 df-en 8200 df-dom 8201 df-sdom 8202 df-fin 8203 df-fsupp 8522 df-sup 8594 df-oi 8661 df-card 9055 df-pnf 10369 df-mnf 10370 df-xr 10371 df-ltxr 10372 df-le 10373 df-sub 10562 df-neg 10563 df-nn 11317 df-2 11380 df-3 11381 df-4 11382 df-5 11383 df-6 11384 df-7 11385 df-8 11386 df-9 11387 df-n0 11585 df-z 11671 df-dec 11788 df-uz 11935 df-fz 12585 df-fzo 12725 df-seq 13060 df-hash 13375 df-struct 16190 df-ndx 16191 df-slot 16192 df-base 16194 df-sets 16195 df-ress 16196 df-plusg 16284 df-mulr 16285 df-sca 16287 df-vsca 16288 df-ip 16289 df-tset 16290 df-ple 16291 df-ds 16293 df-hom 16295 df-cco 16296 df-0g 16421 df-gsum 16422 df-prds 16427 df-pws 16429 df-mre 16565 df-mrc 16566 df-acs 16568 df-mgm 17561 df-sgrp 17603 df-mnd 17614 df-mhm 17654 df-submnd 17655 df-grp 17745 df-minusg 17746 df-sbg 17747 df-mulg 17861 df-subg 17908 df-ghm 17975 df-cntz 18066 df-cmn 18514 df-abl 18515 df-mgp 18810 df-ur 18822 df-ring 18869 df-subrg 19100 df-lmod 19187 df-lss 19255 df-sra 19499 df-rgmod 19500 df-ascl 19641 df-psr 19683 df-mvr 19684 df-mpl 19685 df-opsr 19687 df-psr1 19876 df-vr1 19877 df-ply1 19878 df-dsmm 20405 df-frlm 20420 df-mamu 20519 df-mat 20543 df-mat2pmat 20844 |
This theorem is referenced by: chpdmat 20978 |
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