Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rhmf | Structured version Visualization version GIF version |
Description: A ring homomorphism is a function. (Contributed by Stefan O'Rear, 8-Mar-2015.) |
Ref | Expression |
---|---|
rhmf.b | ⊢ 𝐵 = (Base‘𝑅) |
rhmf.c | ⊢ 𝐶 = (Base‘𝑆) |
Ref | Expression |
---|---|
rhmf | ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝐵⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rhmghm 19599 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | |
2 | rhmf.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | rhmf.c | . . 3 ⊢ 𝐶 = (Base‘𝑆) | |
4 | 2, 3 | ghmf 18480 | . 2 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐵⟶𝐶) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 GrpHom cghm 18473 RingHom crh 19586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-plusg 16681 df-0g 16818 df-mhm 18072 df-ghm 18474 df-mgp 19359 df-ur 19371 df-ring 19418 df-rnghom 19589 |
This theorem is referenced by: rhmf1o 19606 rnrhmsubrg 19686 srngf1o 19744 mulgrhm2 20319 chrrhm 20350 domnchr 20351 znf1o 20370 znidomb 20380 evlslem3 20894 evlslem6 20895 evlslem1 20896 evlseu 20897 mpfconst 20915 mpfproj 20916 mpfsubrg 20917 mpfind 20921 evls1val 21090 evls1sca 21093 evl1val 21099 fveval1fvcl 21103 evl1addd 21111 evl1subd 21112 evl1muld 21113 evl1expd 21115 pf1const 21116 pf1id 21117 pf1subrg 21118 mpfpf1 21121 pf1mpf 21122 pf1ind 21125 ply1remlem 24915 ply1rem 24916 fta1glem1 24918 fta1glem2 24919 fta1g 24920 fta1blem 24921 plypf1 24961 dchrzrhmul 25982 lgsqrlem1 26082 lgsqrlem2 26083 lgsqrlem3 26084 lgseisenlem3 26113 lgseisenlem4 26114 rhmdvdsr 31094 rhmopp 31095 rhmdvd 31097 kerunit 31099 znfermltl 31134 rhmpreimaidl 31175 elrspunidl 31178 rhmimaidl 31181 rhmpreimaprmidl 31199 ply1fermltl 31242 mdetlap 31354 rhmpreimacnlem 31406 pl1cn 31477 zrhunitpreima 31498 elzrhunit 31499 qqhval2lem 31501 qqhf 31506 qqhghm 31508 qqhrhm 31509 qqhnm 31510 selvval2lem4 39810 selvcl 39812 evlsexpval 39855 evlsaddval 39856 evlsmulval 39857 mhphf 39864 idomrootle 40592 elringchom 45106 rhmsscmap2 45111 rhmsscmap 45112 rhmsubcsetclem2 45114 rhmsubcrngclem2 45120 ringcsect 45123 ringcinv 45124 funcringcsetc 45127 funcringcsetcALTV2lem8 45135 funcringcsetcALTV2lem9 45136 elringchomALTV 45141 ringcinvALTV 45148 funcringcsetclem8ALTV 45158 funcringcsetclem9ALTV 45159 zrtermoringc 45162 rhmsubclem4 45181 |
Copyright terms: Public domain | W3C validator |