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Theorem isrnghmmul 44311
Description: A function is a non-unital ring homomorphism iff it preserves both addition and multiplication. (Contributed by AV, 27-Feb-2020.)
Hypotheses
Ref Expression
isrnghmmul.m 𝑀 = (mulGrp‘𝑅)
isrnghmmul.n 𝑁 = (mulGrp‘𝑆)
Assertion
Ref Expression
isrnghmmul (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁))))

Proof of Theorem isrnghmmul
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2821 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2821 . . 3 (.r𝑅) = (.r𝑅)
3 eqid 2821 . . 3 (.r𝑆) = (.r𝑆)
41, 2, 3isrnghm 44310 . 2 (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥)(.r𝑆)(𝐹𝑦)))))
5 isrnghmmul.m . . . . . . . . . . 11 𝑀 = (mulGrp‘𝑅)
65rngmgp 44296 . . . . . . . . . 10 (𝑅 ∈ Rng → 𝑀 ∈ Smgrp)
7 sgrpmgm 17884 . . . . . . . . . 10 (𝑀 ∈ Smgrp → 𝑀 ∈ Mgm)
86, 7syl 17 . . . . . . . . 9 (𝑅 ∈ Rng → 𝑀 ∈ Mgm)
9 isrnghmmul.n . . . . . . . . . . 11 𝑁 = (mulGrp‘𝑆)
109rngmgp 44296 . . . . . . . . . 10 (𝑆 ∈ Rng → 𝑁 ∈ Smgrp)
11 sgrpmgm 17884 . . . . . . . . . 10 (𝑁 ∈ Smgrp → 𝑁 ∈ Mgm)
1210, 11syl 17 . . . . . . . . 9 (𝑆 ∈ Rng → 𝑁 ∈ Mgm)
138, 12anim12i 615 . . . . . . . 8 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm))
14 eqid 2821 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
151, 14ghmf 18340 . . . . . . . 8 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
1613, 15anim12i 615 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ 𝐹 ∈ (𝑅 GrpHom 𝑆)) → ((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ 𝐹:(Base‘𝑅)⟶(Base‘𝑆)))
1716biantrurd 536 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ 𝐹 ∈ (𝑅 GrpHom 𝑆)) → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥)(.r𝑆)(𝐹𝑦)) ↔ (((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ 𝐹:(Base‘𝑅)⟶(Base‘𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥)(.r𝑆)(𝐹𝑦)))))
18 anass 472 . . . . . 6 ((((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ 𝐹:(Base‘𝑅)⟶(Base‘𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥)(.r𝑆)(𝐹𝑦))) ↔ ((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥)(.r𝑆)(𝐹𝑦)))))
1917, 18syl6bb 290 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ 𝐹 ∈ (𝑅 GrpHom 𝑆)) → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥)(.r𝑆)(𝐹𝑦)) ↔ ((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥)(.r𝑆)(𝐹𝑦))))))
205, 1mgpbas 19223 . . . . . 6 (Base‘𝑅) = (Base‘𝑀)
219, 14mgpbas 19223 . . . . . 6 (Base‘𝑆) = (Base‘𝑁)
225, 2mgpplusg 19221 . . . . . 6 (.r𝑅) = (+g𝑀)
239, 3mgpplusg 19221 . . . . . 6 (.r𝑆) = (+g𝑁)
2420, 21, 22, 23ismgmhm 44197 . . . . 5 (𝐹 ∈ (𝑀 MgmHom 𝑁) ↔ ((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥)(.r𝑆)(𝐹𝑦)))))
2519, 24syl6bbr 292 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ 𝐹 ∈ (𝑅 GrpHom 𝑆)) → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥)(.r𝑆)(𝐹𝑦)) ↔ 𝐹 ∈ (𝑀 MgmHom 𝑁)))
2625pm5.32da 582 . . 3 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥)(.r𝑆)(𝐹𝑦))) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁))))
2726pm5.32i 578 . 2 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥)(.r𝑆)(𝐹𝑦)))) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁))))
284, 27bitri 278 1 (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3126  wf 6324  cfv 6328  (class class class)co 7130  Basecbs 16461  .rcmulr 16544  Mgmcmgm 17828  Smgrpcsgrp 17878   GrpHom cghm 18333  mulGrpcmgp 19217   MgmHom cmgmhm 44191  Rngcrng 44292   RngHomo crngh 44303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-map 8383  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-ndx 16464  df-slot 16465  df-base 16467  df-sets 16468  df-plusg 16556  df-sgrp 17879  df-ghm 18334  df-abl 18887  df-mgp 19218  df-mgmhm 44193  df-rng0 44293  df-rnghomo 44305
This theorem is referenced by:  rnghmmgmhm  44312  rnghmval2  44313  rnghmf1o  44321  rnghmco  44325  idrnghm  44326  c0rnghm  44331  rhmisrnghm  44338
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