![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > m2cpmmhm | Structured version Visualization version GIF version |
Description: The transformation of matrices into constant polynomial matrices is a homomorphism of multiplicative monoids. (Contributed by AV, 18-Nov-2019.) |
Ref | Expression |
---|---|
m2cpm.s | β’ π = (π ConstPolyMat π ) |
m2cpm.t | β’ π = (π matToPolyMat π ) |
m2cpm.a | β’ π΄ = (π Mat π ) |
m2cpm.b | β’ π΅ = (Baseβπ΄) |
m2cpmghm.p | β’ π = (Poly1βπ ) |
m2cpmghm.c | β’ πΆ = (π Mat π) |
m2cpmghm.u | β’ π = (πΆ βΎs π) |
Ref | Expression |
---|---|
m2cpmmhm | β’ ((π β Fin β§ π β CRing) β π β ((mulGrpβπ΄) MndHom (mulGrpβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | m2cpm.t | . . 3 β’ π = (π matToPolyMat π ) | |
2 | m2cpm.a | . . 3 β’ π΄ = (π Mat π ) | |
3 | m2cpm.b | . . 3 β’ π΅ = (Baseβπ΄) | |
4 | m2cpmghm.p | . . 3 β’ π = (Poly1βπ ) | |
5 | m2cpmghm.c | . . 3 β’ πΆ = (π Mat π) | |
6 | eqid 2728 | . . 3 β’ (BaseβπΆ) = (BaseβπΆ) | |
7 | 1, 2, 3, 4, 5, 6 | mat2pmatmhm 22663 | . 2 β’ ((π β Fin β§ π β CRing) β π β ((mulGrpβπ΄) MndHom (mulGrpβπΆ))) |
8 | crngring 20199 | . . . . 5 β’ (π β CRing β π β Ring) | |
9 | 8 | anim2i 615 | . . . 4 β’ ((π β Fin β§ π β CRing) β (π β Fin β§ π β Ring)) |
10 | m2cpm.s | . . . . 5 β’ π = (π ConstPolyMat π ) | |
11 | 10, 4, 5 | cpmatsrgpmat 22651 | . . . 4 β’ ((π β Fin β§ π β Ring) β π β (SubRingβπΆ)) |
12 | eqid 2728 | . . . . 5 β’ (mulGrpβπΆ) = (mulGrpβπΆ) | |
13 | 12 | subrgsubm 20538 | . . . 4 β’ (π β (SubRingβπΆ) β π β (SubMndβ(mulGrpβπΆ))) |
14 | 9, 11, 13 | 3syl 18 | . . 3 β’ ((π β Fin β§ π β CRing) β π β (SubMndβ(mulGrpβπΆ))) |
15 | 10, 1, 2, 3 | m2cpmf 22672 | . . . 4 β’ ((π β Fin β§ π β Ring) β π:π΅βΆπ) |
16 | frn 6734 | . . . 4 β’ (π:π΅βΆπ β ran π β π) | |
17 | 9, 15, 16 | 3syl 18 | . . 3 β’ ((π β Fin β§ π β CRing) β ran π β π) |
18 | 5 | ovexi 7460 | . . . . . 6 β’ πΆ β V |
19 | 10 | ovexi 7460 | . . . . . 6 β’ π β V |
20 | m2cpmghm.u | . . . . . . 7 β’ π = (πΆ βΎs π) | |
21 | 20, 12 | mgpress 20103 | . . . . . 6 β’ ((πΆ β V β§ π β V) β ((mulGrpβπΆ) βΎs π) = (mulGrpβπ)) |
22 | 18, 19, 21 | mp2an 690 | . . . . 5 β’ ((mulGrpβπΆ) βΎs π) = (mulGrpβπ) |
23 | 22 | eqcomi 2737 | . . . 4 β’ (mulGrpβπ) = ((mulGrpβπΆ) βΎs π) |
24 | 23 | resmhm2b 18788 | . . 3 β’ ((π β (SubMndβ(mulGrpβπΆ)) β§ ran π β π) β (π β ((mulGrpβπ΄) MndHom (mulGrpβπΆ)) β π β ((mulGrpβπ΄) MndHom (mulGrpβπ)))) |
25 | 14, 17, 24 | syl2anc 582 | . 2 β’ ((π β Fin β§ π β CRing) β (π β ((mulGrpβπ΄) MndHom (mulGrpβπΆ)) β π β ((mulGrpβπ΄) MndHom (mulGrpβπ)))) |
26 | 7, 25 | mpbid 231 | 1 β’ ((π β Fin β§ π β CRing) β π β ((mulGrpβπ΄) MndHom (mulGrpβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3473 β wss 3949 ran crn 5683 βΆwf 6549 βcfv 6553 (class class class)co 7426 Fincfn 8972 Basecbs 17189 βΎs cress 17218 MndHom cmhm 18747 SubMndcsubmnd 18748 mulGrpcmgp 20088 Ringcrg 20187 CRingccrg 20188 SubRingcsubrg 20520 Poly1cpl1 22114 Mat cmat 22335 ConstPolyMat ccpmat 22633 matToPolyMat cmat2pmat 22634 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-ofr 7693 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-pm 8856 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-sup 9475 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-fz 13527 df-fzo 13670 df-seq 14009 df-hash 14332 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-hom 17266 df-cco 17267 df-0g 17432 df-gsum 17433 df-prds 17438 df-pws 17440 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19038 df-subg 19092 df-ghm 19182 df-cntz 19282 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-srg 20141 df-ring 20189 df-cring 20190 df-rhm 20425 df-subrng 20497 df-subrg 20522 df-lmod 20759 df-lss 20830 df-sra 21072 df-rgmod 21073 df-dsmm 21680 df-frlm 21695 df-assa 21801 df-ascl 21803 df-psr 21856 df-mvr 21857 df-mpl 21858 df-opsr 21860 df-psr1 22117 df-vr1 22118 df-ply1 22119 df-coe1 22120 df-mamu 22319 df-mat 22336 df-cpmat 22636 df-mat2pmat 22637 |
This theorem is referenced by: m2cpmrhm 22676 |
Copyright terms: Public domain | W3C validator |