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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 19088 | . 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 18022 | . 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 1656 ∈ wcel 2164 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 GrpHom cghm 18015 RingHom crh 19075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-plusg 16325 df-0g 16462 df-mhm 17695 df-ghm 18016 df-mgp 18851 df-ur 18863 df-ring 18910 df-rnghom 19078 |
This theorem is referenced by: rhmf1o 19095 kerf1hrm 19106 srngf1o 19217 evlslem6 19880 evlslem3 19881 evlslem1 19882 evlseu 19883 mpfconst 19897 mpfproj 19898 mpfsubrg 19899 mpfind 19903 evls1val 20052 evls1sca 20055 evl1val 20060 fveval1fvcl 20064 evl1addd 20072 evl1subd 20073 evl1muld 20074 evl1expd 20076 pf1const 20077 pf1id 20078 pf1subrg 20079 mpfpf1 20082 pf1mpf 20083 pf1ind 20086 mulgrhm2 20214 chrrhm 20246 domnchr 20247 znf1o 20266 znidomb 20276 ply1remlem 24328 ply1rem 24329 fta1glem1 24331 fta1glem2 24332 fta1g 24333 fta1blem 24334 plypf1 24374 dchrzrhmul 25391 lgsqrlem1 25491 lgsqrlem2 25492 lgsqrlem3 25493 lgseisenlem3 25522 lgseisenlem4 25523 rhmdvdsr 30359 rhmopp 30360 rhmdvd 30362 kerunit 30364 mdetlap 30439 pl1cn 30542 zrhunitpreima 30563 elzrhunit 30564 qqhval2lem 30566 qqhf 30571 qqhghm 30573 qqhrhm 30574 qqhnm 30575 idomrootle 38611 elringchom 42875 rhmsscmap2 42880 rhmsscmap 42881 rhmsubcsetclem2 42883 rhmsubcrngclem2 42889 ringcsect 42892 ringcinv 42893 funcringcsetc 42896 funcringcsetcALTV2lem8 42904 funcringcsetcALTV2lem9 42905 elringchomALTV 42910 ringcinvALTV 42917 funcringcsetclem8ALTV 42927 funcringcsetclem9ALTV 42928 zrtermoringc 42931 rhmsubclem4 42950 |
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