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Mirrors > Home > MPE Home > Th. List > pm2mpghmlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for pm2mpghm 20998. (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.) |
Ref | Expression |
---|---|
pm2mpfo.p | ⊢ 𝑃 = (Poly1‘𝑅) |
pm2mpfo.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
pm2mpfo.b | ⊢ 𝐵 = (Base‘𝐶) |
pm2mpfo.m | ⊢ ∗ = ( ·𝑠 ‘𝑄) |
pm2mpfo.e | ⊢ ↑ = (.g‘(mulGrp‘𝑄)) |
pm2mpfo.x | ⊢ 𝑋 = (var1‘𝐴) |
pm2mpfo.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
pm2mpfo.q | ⊢ 𝑄 = (Poly1‘𝐴) |
Ref | Expression |
---|---|
pm2mpghmlem2 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) finSupp (0g‘𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ex 11632 | . . 3 ⊢ ℕ0 ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ℕ0 ∈ V) |
3 | pm2mpfo.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
4 | 3 | matring 20623 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
5 | 4 | 3adant3 1166 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring) |
6 | pm2mpfo.q | . . . 4 ⊢ 𝑄 = (Poly1‘𝐴) | |
7 | 6 | ply1lmod 19989 | . . 3 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod) |
8 | 5, 7 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑄 ∈ LMod) |
9 | 6 | ply1sca 19990 | . . 3 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄)) |
10 | 5, 9 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴 = (Scalar‘𝑄)) |
11 | eqid 2825 | . 2 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
12 | simpl2 1248 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring) | |
13 | simpl3 1250 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ 𝐵) | |
14 | simpr 479 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
15 | pm2mpfo.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
16 | pm2mpfo.c | . . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
17 | pm2mpfo.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
18 | eqid 2825 | . . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
19 | 15, 16, 17, 3, 18 | decpmatcl 20949 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) → (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴)) |
20 | 12, 13, 14, 19 | syl3anc 1494 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴)) |
21 | pm2mpfo.x | . . . 4 ⊢ 𝑋 = (var1‘𝐴) | |
22 | eqid 2825 | . . . 4 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄) | |
23 | pm2mpfo.e | . . . 4 ⊢ ↑ = (.g‘(mulGrp‘𝑄)) | |
24 | 6, 21, 22, 23, 11 | ply1moncl 20008 | . . 3 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ (Base‘𝑄)) |
25 | 5, 24 | sylan 575 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ (Base‘𝑄)) |
26 | eqid 2825 | . 2 ⊢ (0g‘𝑄) = (0g‘𝑄) | |
27 | eqid 2825 | . 2 ⊢ (0g‘𝐴) = (0g‘𝐴) | |
28 | pm2mpfo.m | . 2 ⊢ ∗ = ( ·𝑠 ‘𝑄) | |
29 | 15, 16, 17, 3, 27 | decpmatfsupp 20951 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑀 decompPMat 𝑘)) finSupp (0g‘𝐴)) |
30 | 29 | 3adant1 1164 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑀 decompPMat 𝑘)) finSupp (0g‘𝐴)) |
31 | 2, 8, 10, 11, 20, 25, 26, 27, 28, 30 | mptscmfsupp0 19291 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) finSupp (0g‘𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 Vcvv 3414 class class class wbr 4875 ↦ cmpt 4954 ‘cfv 6127 (class class class)co 6910 Fincfn 8228 finSupp cfsupp 8550 ℕ0cn0 11625 Basecbs 16229 Scalarcsca 16315 ·𝑠 cvsca 16316 0gc0g 16460 .gcmg 17901 mulGrpcmgp 18850 Ringcrg 18908 LModclmod 19226 var1cv1 19913 Poly1cpl1 19914 Mat cmat 20587 decompPMat cdecpmat 20944 |
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-fal 1670 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-coe1 19920 df-dsmm 20446 df-frlm 20461 df-mamu 20564 df-mat 20588 df-decpmat 20945 |
This theorem is referenced by: pm2mpghm 20998 pm2mpmhmlem2 21001 |
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