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Mirrors > Home > MPE Home > Th. List > pm2mpmhm | Structured version Visualization version GIF version |
Description: The transformation of polynomial matrices into polynomials over matrices is a homomorphism of multiplicative monoids. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.) |
Ref | Expression |
---|---|
pm2mpmhm.p | โข ๐ = (Poly1โ๐ ) |
pm2mpmhm.c | โข ๐ถ = (๐ Mat ๐) |
pm2mpmhm.a | โข ๐ด = (๐ Mat ๐ ) |
pm2mpmhm.q | โข ๐ = (Poly1โ๐ด) |
pm2mpmhm.t | โข ๐ = (๐ pMatToMatPoly ๐ ) |
Ref | Expression |
---|---|
pm2mpmhm | โข ((๐ โ Fin โง ๐ โ Ring) โ ๐ โ ((mulGrpโ๐ถ) MndHom (mulGrpโ๐))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2mpmhm.p | . . . 4 โข ๐ = (Poly1โ๐ ) | |
2 | pm2mpmhm.c | . . . 4 โข ๐ถ = (๐ Mat ๐) | |
3 | 1, 2 | pmatring 22612 | . . 3 โข ((๐ โ Fin โง ๐ โ Ring) โ ๐ถ โ Ring) |
4 | eqid 2725 | . . . 4 โข (mulGrpโ๐ถ) = (mulGrpโ๐ถ) | |
5 | 4 | ringmgp 20183 | . . 3 โข (๐ถ โ Ring โ (mulGrpโ๐ถ) โ Mnd) |
6 | 3, 5 | syl 17 | . 2 โข ((๐ โ Fin โง ๐ โ Ring) โ (mulGrpโ๐ถ) โ Mnd) |
7 | pm2mpmhm.a | . . . 4 โข ๐ด = (๐ Mat ๐ ) | |
8 | 7 | matring 22363 | . . 3 โข ((๐ โ Fin โง ๐ โ Ring) โ ๐ด โ Ring) |
9 | pm2mpmhm.q | . . . 4 โข ๐ = (Poly1โ๐ด) | |
10 | 9 | ply1ring 22175 | . . 3 โข (๐ด โ Ring โ ๐ โ Ring) |
11 | eqid 2725 | . . . 4 โข (mulGrpโ๐) = (mulGrpโ๐) | |
12 | 11 | ringmgp 20183 | . . 3 โข (๐ โ Ring โ (mulGrpโ๐) โ Mnd) |
13 | 8, 10, 12 | 3syl 18 | . 2 โข ((๐ โ Fin โง ๐ โ Ring) โ (mulGrpโ๐) โ Mnd) |
14 | eqid 2725 | . . . . . 6 โข (Baseโ๐ถ) = (Baseโ๐ถ) | |
15 | 4, 14 | mgpbas 20084 | . . . . 5 โข (Baseโ๐ถ) = (Baseโ(mulGrpโ๐ถ)) |
16 | 15 | eqcomi 2734 | . . . 4 โข (Baseโ(mulGrpโ๐ถ)) = (Baseโ๐ถ) |
17 | eqid 2725 | . . . 4 โข ( ยท๐ โ๐) = ( ยท๐ โ๐) | |
18 | eqid 2725 | . . . 4 โข (.gโ(mulGrpโ๐)) = (.gโ(mulGrpโ๐)) | |
19 | eqid 2725 | . . . 4 โข (var1โ๐ด) = (var1โ๐ด) | |
20 | pm2mpmhm.t | . . . 4 โข ๐ = (๐ pMatToMatPoly ๐ ) | |
21 | eqid 2725 | . . . . . 6 โข (Baseโ๐) = (Baseโ๐) | |
22 | 11, 21 | mgpbas 20084 | . . . . 5 โข (Baseโ๐) = (Baseโ(mulGrpโ๐)) |
23 | 22 | eqcomi 2734 | . . . 4 โข (Baseโ(mulGrpโ๐)) = (Baseโ๐) |
24 | 1, 2, 16, 17, 18, 19, 7, 9, 20, 23 | pm2mpf 22718 | . . 3 โข ((๐ โ Fin โง ๐ โ Ring) โ ๐:(Baseโ(mulGrpโ๐ถ))โถ(Baseโ(mulGrpโ๐))) |
25 | 1, 2, 7, 9, 20, 16 | pm2mpmhmlem2 22739 | . . 3 โข ((๐ โ Fin โง ๐ โ Ring) โ โ๐ฅ โ (Baseโ(mulGrpโ๐ถ))โ๐ฆ โ (Baseโ(mulGrpโ๐ถ))(๐โ(๐ฅ(.rโ๐ถ)๐ฆ)) = ((๐โ๐ฅ)(.rโ๐)(๐โ๐ฆ))) |
26 | 1, 2, 14, 17, 18, 19, 7, 9, 20 | idpm2idmp 22721 | . . 3 โข ((๐ โ Fin โง ๐ โ Ring) โ (๐โ(1rโ๐ถ)) = (1rโ๐)) |
27 | 24, 25, 26 | 3jca 1125 | . 2 โข ((๐ โ Fin โง ๐ โ Ring) โ (๐:(Baseโ(mulGrpโ๐ถ))โถ(Baseโ(mulGrpโ๐)) โง โ๐ฅ โ (Baseโ(mulGrpโ๐ถ))โ๐ฆ โ (Baseโ(mulGrpโ๐ถ))(๐โ(๐ฅ(.rโ๐ถ)๐ฆ)) = ((๐โ๐ฅ)(.rโ๐)(๐โ๐ฆ)) โง (๐โ(1rโ๐ถ)) = (1rโ๐))) |
28 | eqid 2725 | . . 3 โข (Baseโ(mulGrpโ๐ถ)) = (Baseโ(mulGrpโ๐ถ)) | |
29 | eqid 2725 | . . 3 โข (Baseโ(mulGrpโ๐)) = (Baseโ(mulGrpโ๐)) | |
30 | eqid 2725 | . . . 4 โข (.rโ๐ถ) = (.rโ๐ถ) | |
31 | 4, 30 | mgpplusg 20082 | . . 3 โข (.rโ๐ถ) = (+gโ(mulGrpโ๐ถ)) |
32 | eqid 2725 | . . . 4 โข (.rโ๐) = (.rโ๐) | |
33 | 11, 32 | mgpplusg 20082 | . . 3 โข (.rโ๐) = (+gโ(mulGrpโ๐)) |
34 | eqid 2725 | . . . 4 โข (1rโ๐ถ) = (1rโ๐ถ) | |
35 | 4, 34 | ringidval 20127 | . . 3 โข (1rโ๐ถ) = (0gโ(mulGrpโ๐ถ)) |
36 | eqid 2725 | . . . 4 โข (1rโ๐) = (1rโ๐) | |
37 | 11, 36 | ringidval 20127 | . . 3 โข (1rโ๐) = (0gโ(mulGrpโ๐)) |
38 | 28, 29, 31, 33, 35, 37 | ismhm 18741 | . 2 โข (๐ โ ((mulGrpโ๐ถ) MndHom (mulGrpโ๐)) โ (((mulGrpโ๐ถ) โ Mnd โง (mulGrpโ๐) โ Mnd) โง (๐:(Baseโ(mulGrpโ๐ถ))โถ(Baseโ(mulGrpโ๐)) โง โ๐ฅ โ (Baseโ(mulGrpโ๐ถ))โ๐ฆ โ (Baseโ(mulGrpโ๐ถ))(๐โ(๐ฅ(.rโ๐ถ)๐ฆ)) = ((๐โ๐ฅ)(.rโ๐)(๐โ๐ฆ)) โง (๐โ(1rโ๐ถ)) = (1rโ๐)))) |
39 | 6, 13, 27, 38 | syl21anbrc 1341 | 1 โข ((๐ โ Fin โง ๐ โ Ring) โ ๐ โ ((mulGrpโ๐ถ) MndHom (mulGrpโ๐))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 394 โง w3a 1084 = wceq 1533 โ wcel 2098 โwral 3051 โถwf 6539 โcfv 6543 (class class class)co 7416 Fincfn 8962 Basecbs 17179 .rcmulr 17233 ยท๐ cvsca 17236 Mndcmnd 18693 MndHom cmhm 18737 .gcmg 19027 mulGrpcmgp 20078 1rcur 20125 Ringcrg 20177 var1cv1 22103 Poly1cpl1 22104 Mat cmat 22325 pMatToMatPoly cpm2mp 22712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-ofr 7683 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-ghm 19172 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-srg 20131 df-ring 20179 df-subrng 20487 df-subrg 20512 df-lmod 20749 df-lss 20820 df-sra 21062 df-rgmod 21063 df-dsmm 21670 df-frlm 21685 df-ascl 21793 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-psr1 22107 df-vr1 22108 df-ply1 22109 df-coe1 22110 df-mamu 22309 df-mat 22326 df-decpmat 22683 df-pm2mp 22713 |
This theorem is referenced by: pm2mprhm 22741 |
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