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| Mirrors > Home > MPE Home > Th. List > rhmghm | Structured version Visualization version GIF version | ||
| Description: A ring homomorphism is an additive group homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
| Ref | Expression |
|---|---|
| rhmghm | ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
| 2 | eqid 2737 | . . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
| 3 | 1, 2 | isrhm 20478 | . . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))) |
| 4 | 3 | simprbi 496 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)))) |
| 5 | 4 | simpld 494 | 1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 MndHom cmhm 18794 GrpHom cghm 19230 mulGrpcmgp 20137 Ringcrg 20230 RingHom crh 20469 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-0g 17486 df-mhm 18796 df-ghm 19231 df-mgp 20138 df-ur 20179 df-ring 20232 df-rhm 20472 |
| This theorem is referenced by: rhmf 20485 rhmf1o 20491 rimgim 20497 rhmco 20501 pwsco2rhm 20503 rhmopp 20509 nrhmzr 20537 rhmimasubrng 20566 resrhm 20601 rhmeql 20603 rhmima 20604 imadrhmcl 20798 srngadd 20852 srng0 20855 rhmpreimaidl 21287 rhmqusnsg 21295 mulgrhm2 21489 zrh0 21524 fermltlchr 21544 chrrhm 21546 zndvds0 21569 zzngim 21571 cygznlem3 21588 zrhpsgnodpm 21610 mplind 22094 evlslem3 22104 evlslem6 22105 evlslem1 22106 evlsgsumadd 22115 mpfind 22131 evls1gsumadd 22328 evl1addd 22345 evl1subd 22346 evls1maplmhm 22381 rhmcomulmpl 22386 rhmmpl 22387 rhmply1vr1 22391 rhmply1vsca 22392 ply1rem 26205 plypf1 26251 znfermltl 33394 rhmquskerlem 33453 rhmqusker 33454 rhmimaidl 33460 algextdeglem4 33761 zrhf1ker 33974 zrhneg 33979 zrhcntr 33980 qqhghm 33989 qqhrhm 33990 rhmzrhval 41971 fldhmf1 42091 aks6d1c1p2 42110 aks6d1c1p3 42111 aks6d1c5lem1 42137 aks6d1c5lem2 42139 rhmqusspan 42186 aks5lem2 42188 aks5lem3a 42190 ricdrng1 42538 rhmcomulpsr 42561 rhmpsr 42562 evlsaddval 42578 evladdval 42585 selvcllem4 42591 selvvvval 42595 selvadd 42598 selvmul 42599 |
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