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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 20378 | . 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 19137 | . 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 1533 ∈ wcel 2098 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 Basecbs 17145 GrpHom cghm 19130 RingHom crh 20363 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-plusg 17211 df-0g 17388 df-mhm 18705 df-ghm 19131 df-mgp 20032 df-ur 20079 df-ring 20132 df-rhm 20366 |
This theorem is referenced by: rhmf1o 20385 rhmdvdsr 20402 rhmopp 20403 rnrhmsubrg 20499 elringchom 20541 rhmsscmap2 20546 rhmsscmap 20547 rhmsubcsetclem2 20549 rhmsubcrngclem2 20555 ringcsect 20558 ringcinv 20559 funcringcsetc 20562 zrtermoringc 20563 rhmsubclem4 20576 imadrhmcl 20640 srngf1o 20689 mulgrhm2 21335 fermltlchr 21390 chrrhm 21392 domnchr 21393 znf1o 21416 znidomb 21426 evlslem3 21955 evlslem6 21956 evlslem1 21957 evlseu 21958 mpfconst 21976 mpfproj 21977 mpfsubrg 21978 mpfind 21982 ply1fermltlchr 22155 evls1val 22163 evls1sca 22166 evl1val 22172 fveval1fvcl 22176 evl1addd 22184 evl1subd 22185 evl1muld 22186 evl1expd 22188 pf1const 22189 pf1id 22190 pf1subrg 22191 mpfpf1 22194 pf1mpf 22195 pf1ind 22198 ply1remlem 26022 ply1rem 26023 fta1glem1 26025 fta1glem2 26026 fta1g 26027 fta1blem 26028 plypf1 26068 dchrzrhmul 27098 lgsqrlem1 27198 lgsqrlem2 27199 lgsqrlem3 27200 lgseisenlem3 27229 lgseisenlem4 27230 rndrhmcl 32865 rhmdvd 32905 kerunit 32906 znfermltl 32951 rhmpreimaidl 33009 elrspunidl 33018 rhmimaidl 33022 rhmpreimaprmidl 33042 evls1fn 33110 evls1dm 33111 evls1fvf 33112 evls1expd 33114 evls1fpws 33116 ressply1evl 33117 elirng 33233 irngss 33234 irngnzply1lem 33237 irngnzply1 33238 mdetlap 33304 rhmpreimacnlem 33356 pl1cn 33427 zrhunitpreima 33450 elzrhunit 33451 qqhval2lem 33453 qqhf 33458 qqhghm 33460 qqhrhm 33461 qqhnm 33462 fldhmf1 41452 imacrhmcl 41584 rimcnv 41587 rhmcomulmpl 41617 rhmmpl 41618 evlscl 41623 evlsexpval 41632 evlsaddval 41633 evlsmulval 41634 evlcl 41637 evladdval 41640 evlmulval 41641 selvcl 41648 idomrootle 42451 funcringcsetcALTV2lem8 47185 funcringcsetcALTV2lem9 47186 elringchomALTV 47191 ringcinvALTV 47198 funcringcsetclem8ALTV 47208 funcringcsetclem9ALTV 47209 |
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