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Mirrors > Home > MPE Home > Th. List > pm2mpghmlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for pm2mpghm . (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) |
pm2mpfo.l | ⊢ 𝐿 = (Base‘𝑄) |
Ref | Expression |
---|---|
pm2mpghmlem1 | ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑀 decompPMat 𝐾) ∗ (𝐾 ↑ 𝑋)) ∈ 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2mpfo.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | 1 | matring 21914 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
3 | 2 | 3adant3 1133 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring) |
4 | 3 | adantr 482 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ Ring) |
5 | simpl2 1193 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Ring) | |
6 | simpl3 1194 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑀 ∈ 𝐵) | |
7 | simpr 486 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0) | |
8 | pm2mpfo.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
9 | pm2mpfo.c | . . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
10 | pm2mpfo.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
11 | eqid 2733 | . . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
12 | 8, 9, 10, 1, 11 | decpmatcl 22238 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) ∈ (Base‘𝐴)) |
13 | 5, 6, 7, 12 | syl3anc 1372 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) ∈ (Base‘𝐴)) |
14 | pm2mpfo.q | . . 3 ⊢ 𝑄 = (Poly1‘𝐴) | |
15 | pm2mpfo.x | . . 3 ⊢ 𝑋 = (var1‘𝐴) | |
16 | pm2mpfo.m | . . 3 ⊢ ∗ = ( ·𝑠 ‘𝑄) | |
17 | eqid 2733 | . . 3 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄) | |
18 | pm2mpfo.e | . . 3 ⊢ ↑ = (.g‘(mulGrp‘𝑄)) | |
19 | pm2mpfo.l | . . 3 ⊢ 𝐿 = (Base‘𝑄) | |
20 | 11, 14, 15, 16, 17, 18, 19 | ply1tmcl 21765 | . 2 ⊢ ((𝐴 ∈ Ring ∧ (𝑀 decompPMat 𝐾) ∈ (Base‘𝐴) ∧ 𝐾 ∈ ℕ0) → ((𝑀 decompPMat 𝐾) ∗ (𝐾 ↑ 𝑋)) ∈ 𝐿) |
21 | 4, 13, 7, 20 | syl3anc 1372 | 1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑀 decompPMat 𝐾) ∗ (𝐾 ↑ 𝑋)) ∈ 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6535 (class class class)co 7396 Fincfn 8927 ℕ0cn0 12459 Basecbs 17131 ·𝑠 cvsca 17188 .gcmg 18935 mulGrpcmgp 19970 Ringcrg 20038 var1cv1 21669 Poly1cpl1 21670 Mat cmat 21876 decompPMat cdecpmat 22233 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7657 df-ofr 7658 df-om 7843 df-1st 7962 df-2nd 7963 df-supp 8134 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9350 df-sup 9424 df-oi 9492 df-card 9921 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-fz 13472 df-fzo 13615 df-seq 13954 df-hash 14278 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-sca 17200 df-vsca 17201 df-ip 17202 df-tset 17203 df-ple 17204 df-ds 17206 df-hom 17208 df-cco 17209 df-0g 17374 df-gsum 17375 df-prds 17380 df-pws 17382 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-mhm 18658 df-submnd 18659 df-grp 18809 df-minusg 18810 df-sbg 18811 df-mulg 18936 df-subg 18988 df-ghm 19075 df-cntz 19166 df-cmn 19634 df-abl 19635 df-mgp 19971 df-ur 19988 df-ring 20040 df-subrg 20338 df-lmod 20450 df-lss 20520 df-sra 20762 df-rgmod 20763 df-dsmm 21260 df-frlm 21275 df-psr 21433 df-mvr 21434 df-mpl 21435 df-opsr 21437 df-psr1 21673 df-vr1 21674 df-ply1 21675 df-coe1 21676 df-mamu 21855 df-mat 21877 df-decpmat 22234 |
This theorem is referenced by: pm2mpghm 22287 pm2mpmhmlem2 22290 |
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