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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 2821 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | eqid 2821 | . . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
3 | 1, 2 | isrhm 19473 | . . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))) |
4 | 3 | simprbi 499 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)))) |
5 | 4 | simpld 497 | 1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 MndHom cmhm 17954 GrpHom cghm 18355 mulGrpcmgp 19239 Ringcrg 19297 RingHom crh 19464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-0g 16715 df-mhm 17956 df-ghm 18356 df-mgp 19240 df-ur 19252 df-ring 19299 df-rnghom 19467 |
This theorem is referenced by: rhmf 19478 rhmf1o 19484 rimgim 19488 rhmco 19489 pwsco2rhm 19491 f1rhm0to0OLD 19493 f1rhm0to0ALT 19494 kerf1hrmOLD 19498 resrhm 19564 rhmeql 19565 rhmima 19566 srngadd 19628 srng0 19631 mplind 20282 evlslem3 20293 evlslem6 20294 evlslem1 20295 evlsgsumadd 20304 mpfind 20320 evls1gsumadd 20487 evl1addd 20504 evl1subd 20505 mulgrhm2 20646 zrh0 20661 chrrhm 20678 zndvds0 20697 zzngim 20699 cygznlem3 20716 zrhpsgnodpm 20736 ply1rem 24757 plypf1 24802 rhmopp 30892 zrhf1ker 31216 qqhghm 31229 qqhrhm 31230 nrhmzr 44193 |
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