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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 20153 | . . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))) |
4 | 3 | simprbi 498 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)))) |
5 | 4 | simpld 496 | 1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ‘cfv 6497 (class class class)co 7358 MndHom cmhm 18600 GrpHom cghm 19006 mulGrpcmgp 19897 Ringcrg 19965 RingHom crh 20144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-0g 17324 df-mhm 18602 df-ghm 19007 df-mgp 19898 df-ur 19915 df-ring 19967 df-rnghom 20147 |
This theorem is referenced by: rhmf 20159 rhmf1o 20165 rimgim 20171 rhmco 20172 pwsco2rhm 20174 f1rhm0to0ALT 20176 rhmopp 20183 resrhm 20254 rhmeql 20255 rhmima 20256 srngadd 20319 srng0 20322 mulgrhm2 20902 zrh0 20917 chrrhm 20937 zndvds0 20960 zzngim 20962 cygznlem3 20979 zrhpsgnodpm 20999 mplind 21481 evlslem3 21493 evlslem6 21494 evlslem1 21495 evlsgsumadd 21504 mpfind 21520 evls1gsumadd 21693 evl1addd 21710 evl1subd 21711 ply1rem 25531 plypf1 25576 fermltlchr 32157 znfermltl 32158 rhmpreimaidl 32203 rhmimaidl 32209 zrhf1ker 32559 qqhghm 32572 qqhrhm 32573 fldhmf1 40550 imadrhmcl 40719 ricdrng1 40720 rhmcomulmpl 40743 rhmmpl 40744 evlsaddval 40753 evladdval 40756 selvcllem4 40762 selvadd 40766 selvmul 40767 nrhmzr 46178 |
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