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Mirrors > Home > MPE Home > Th. List > pm2mprhm | Structured version Visualization version GIF version |
Description: The transformation of polynomial matrices into polynomials over matrices is a ring homomorphism. (Contributed by AV, 22-Oct-2019.) |
Ref | Expression |
---|---|
pm2mpmhm.p | ⊢ 𝑃 = (Poly1‘𝑅) |
pm2mpmhm.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
pm2mpmhm.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
pm2mpmhm.q | ⊢ 𝑄 = (Poly1‘𝐴) |
pm2mpmhm.t | ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) |
Ref | Expression |
---|---|
pm2mprhm | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingHom 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2mpmhm.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | pm2mpmhm.c | . . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
3 | 1, 2 | pmatring 21005 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring) |
4 | pm2mpmhm.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
5 | 4 | matring 20756 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
6 | pm2mpmhm.q | . . . 4 ⊢ 𝑄 = (Poly1‘𝐴) | |
7 | 6 | ply1ring 20119 | . . 3 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring) |
8 | 5, 7 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring) |
9 | eqid 2779 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
10 | eqid 2779 | . . . 4 ⊢ ( ·𝑠 ‘𝑄) = ( ·𝑠 ‘𝑄) | |
11 | eqid 2779 | . . . 4 ⊢ (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄)) | |
12 | eqid 2779 | . . . 4 ⊢ (var1‘𝐴) = (var1‘𝐴) | |
13 | eqid 2779 | . . . 4 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
14 | pm2mpmhm.t | . . . 4 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) | |
15 | 1, 2, 9, 10, 11, 12, 4, 6, 13, 14 | pm2mpghm 21128 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 GrpHom 𝑄)) |
16 | 1, 2, 4, 6, 14 | pm2mpmhm 21132 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄))) |
17 | 15, 16 | jca 504 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇 ∈ (𝐶 GrpHom 𝑄) ∧ 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄)))) |
18 | eqid 2779 | . . 3 ⊢ (mulGrp‘𝐶) = (mulGrp‘𝐶) | |
19 | eqid 2779 | . . 3 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄) | |
20 | 18, 19 | isrhm 19196 | . 2 ⊢ (𝑇 ∈ (𝐶 RingHom 𝑄) ↔ ((𝐶 ∈ Ring ∧ 𝑄 ∈ Ring) ∧ (𝑇 ∈ (𝐶 GrpHom 𝑄) ∧ 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄))))) |
21 | 3, 8, 17, 20 | syl21anbrc 1324 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingHom 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ‘cfv 6188 (class class class)co 6976 Fincfn 8306 Basecbs 16339 ·𝑠 cvsca 16425 MndHom cmhm 17801 .gcmg 18011 GrpHom cghm 18126 mulGrpcmgp 18962 Ringcrg 19020 RingHom crh 19187 var1cv1 20047 Poly1cpl1 20048 Mat cmat 20720 pMatToMatPoly cpm2mp 21104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-ot 4450 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-ofr 7228 df-om 7397 df-1st 7501 df-2nd 7502 df-supp 7634 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-ixp 8260 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-fsupp 8629 df-sup 8701 df-oi 8769 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-fz 12709 df-fzo 12850 df-seq 13185 df-hash 13506 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-hom 16445 df-cco 16446 df-0g 16571 df-gsum 16572 df-prds 16577 df-pws 16579 df-mre 16715 df-mrc 16716 df-acs 16718 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-mhm 17803 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-mulg 18012 df-subg 18060 df-ghm 18127 df-cntz 18218 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-srg 18979 df-ring 19022 df-rnghom 19190 df-subrg 19256 df-lmod 19358 df-lss 19426 df-sra 19666 df-rgmod 19667 df-ascl 19808 df-psr 19850 df-mvr 19851 df-mpl 19852 df-opsr 19854 df-psr1 20051 df-vr1 20052 df-ply1 20053 df-coe1 20054 df-dsmm 20578 df-frlm 20593 df-mamu 20697 df-mat 20721 df-decpmat 21075 df-pm2mp 21105 |
This theorem is referenced by: pm2mprngiso 21134 |
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