![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rhmmul | Structured version Visualization version GIF version |
Description: A homomorphism of rings preserves multiplication. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
rhmmul.x | โข ๐ = (Baseโ๐ ) |
rhmmul.m | โข ยท = (.rโ๐ ) |
rhmmul.n | โข ร = (.rโ๐) |
Ref | Expression |
---|---|
rhmmul | โข ((๐น โ (๐ RingHom ๐) โง ๐ด โ ๐ โง ๐ต โ ๐) โ (๐นโ(๐ด ยท ๐ต)) = ((๐นโ๐ด) ร (๐นโ๐ต))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 โข (mulGrpโ๐ ) = (mulGrpโ๐ ) | |
2 | eqid 2726 | . . 3 โข (mulGrpโ๐) = (mulGrpโ๐) | |
3 | 1, 2 | rhmmhm 20379 | . 2 โข (๐น โ (๐ RingHom ๐) โ ๐น โ ((mulGrpโ๐ ) MndHom (mulGrpโ๐))) |
4 | rhmmul.x | . . . 4 โข ๐ = (Baseโ๐ ) | |
5 | 1, 4 | mgpbas 20043 | . . 3 โข ๐ = (Baseโ(mulGrpโ๐ )) |
6 | rhmmul.m | . . . 4 โข ยท = (.rโ๐ ) | |
7 | 1, 6 | mgpplusg 20041 | . . 3 โข ยท = (+gโ(mulGrpโ๐ )) |
8 | rhmmul.n | . . . 4 โข ร = (.rโ๐) | |
9 | 2, 8 | mgpplusg 20041 | . . 3 โข ร = (+gโ(mulGrpโ๐)) |
10 | 5, 7, 9 | mhmlin 18721 | . 2 โข ((๐น โ ((mulGrpโ๐ ) MndHom (mulGrpโ๐)) โง ๐ด โ ๐ โง ๐ต โ ๐) โ (๐นโ(๐ด ยท ๐ต)) = ((๐นโ๐ด) ร (๐นโ๐ต))) |
11 | 3, 10 | syl3an1 1160 | 1 โข ((๐น โ (๐ RingHom ๐) โง ๐ด โ ๐ โง ๐ต โ ๐) โ (๐นโ(๐ด ยท ๐ต)) = ((๐นโ๐ด) ร (๐นโ๐ต))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง w3a 1084 = wceq 1533 โ wcel 2098 โcfv 6536 (class class class)co 7404 Basecbs 17151 .rcmulr 17205 MndHom cmhm 18709 mulGrpcmgp 20037 RingHom crh 20369 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-0g 17394 df-mhm 18711 df-ghm 19137 df-mgp 20038 df-ur 20085 df-ring 20138 df-rhm 20372 |
This theorem is referenced by: rhmdvdsr 20408 rhmopp 20409 rhmunitinv 20411 srngmul 20699 domnchr 21419 znfld 21451 znidomb 21452 znunit 21454 znrrg 21456 evl1muld 22213 evl1scvarpw 22233 mat2pmatmul 22584 mat2pmatlin 22588 cayhamlem4 22741 ply1rem 26051 fta1glem2 26054 fta1blem 26056 dchrzrhmul 27130 lgsdchr 27239 lgseisenlem3 27261 lgseisenlem4 27262 rhmdvd 32939 kerunit 32940 rhmpreimaidl 33043 rhmquskerlem 33049 rhmimaidl 33056 rhmpreimaprmidl 33076 evls1fpws 33155 mdetpmtr1 33333 mdetpmtr12 33335 qqhghm 33498 qqhrhm 33499 fldhmf1 41469 imacrhmcl 41629 rhmcomulmpl 41662 evlsmulval 41679 evlmulval 41686 |
Copyright terms: Public domain | W3C validator |