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Mirrors > Home > MPE Home > Th. List > pm2mpghmlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for pm2mpghm 22804. (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 12522 | . . 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 22431 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
5 | 4 | 3adant3 1129 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring) |
6 | pm2mpfo.q | . . . 4 ⊢ 𝑄 = (Poly1‘𝐴) | |
7 | 6 | ply1lmod 22235 | . . 3 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod) |
8 | 5, 7 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑄 ∈ LMod) |
9 | 6 | ply1sca 22236 | . . 3 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄)) |
10 | 5, 9 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴 = (Scalar‘𝑄)) |
11 | eqid 2726 | . 2 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
12 | simpl2 1189 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring) | |
13 | simpl3 1190 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ 𝐵) | |
14 | simpr 483 | . . 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 2726 | . . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
19 | 15, 16, 17, 3, 18 | decpmatcl 22755 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) → (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴)) |
20 | 12, 13, 14, 19 | syl3anc 1368 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑀 decompPMat 𝑘) ∈ (Base‘𝐴)) |
21 | pm2mpfo.x | . . . 4 ⊢ 𝑋 = (var1‘𝐴) | |
22 | eqid 2726 | . . . 4 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄) | |
23 | pm2mpfo.e | . . . 4 ⊢ ↑ = (.g‘(mulGrp‘𝑄)) | |
24 | 6, 21, 22, 23, 11 | ply1moncl 22256 | . . 3 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ (Base‘𝑄)) |
25 | 5, 24 | sylan 578 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ (Base‘𝑄)) |
26 | eqid 2726 | . 2 ⊢ (0g‘𝑄) = (0g‘𝑄) | |
27 | eqid 2726 | . 2 ⊢ (0g‘𝐴) = (0g‘𝐴) | |
28 | pm2mpfo.m | . 2 ⊢ ∗ = ( ·𝑠 ‘𝑄) | |
29 | 15, 16, 17, 3, 27 | decpmatfsupp 22757 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑀 decompPMat 𝑘)) finSupp (0g‘𝐴)) |
30 | 29 | 3adant1 1127 | . 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 20897 | 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 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 Vcvv 3463 class class class wbr 5144 ↦ cmpt 5227 ‘cfv 6544 (class class class)co 7414 Fincfn 8964 finSupp cfsupp 9396 ℕ0cn0 12516 Basecbs 17206 Scalarcsca 17262 ·𝑠 cvsca 17263 0gc0g 17447 .gcmg 19055 mulGrpcmgp 20111 Ringcrg 20210 LModclmod 20830 var1cv1 22159 Poly1cpl1 22160 Mat cmat 22393 decompPMat cdecpmat 22750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4907 df-int 4948 df-iun 4996 df-iin 4997 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-ofr 7681 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9397 df-sup 9476 df-oi 9544 df-card 9973 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-7 12324 df-8 12325 df-9 12326 df-n0 12517 df-z 12603 df-dec 12722 df-uz 12867 df-fz 13531 df-fzo 13674 df-seq 14014 df-hash 14341 df-struct 17142 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-ress 17236 df-plusg 17272 df-mulr 17273 df-sca 17275 df-vsca 17276 df-ip 17277 df-tset 17278 df-ple 17279 df-ds 17281 df-hom 17283 df-cco 17284 df-0g 17449 df-gsum 17450 df-prds 17455 df-pws 17457 df-mre 17592 df-mrc 17593 df-acs 17595 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-ghm 19201 df-cntz 19305 df-cmn 19774 df-abl 19775 df-mgp 20112 df-rng 20130 df-ur 20159 df-ring 20212 df-subrng 20522 df-subrg 20547 df-lmod 20832 df-lss 20903 df-sra 21145 df-rgmod 21146 df-dsmm 21724 df-frlm 21739 df-psr 21900 df-mvr 21901 df-mpl 21902 df-opsr 21904 df-psr1 22163 df-vr1 22164 df-ply1 22165 df-coe1 22166 df-mamu 22377 df-mat 22394 df-decpmat 22751 |
This theorem is referenced by: pm2mpghm 22804 pm2mpmhmlem2 22807 |
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