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Theorem rngogrphom 37501
Description: A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.)
Hypotheses
Ref Expression
rnggrphom.1 𝐺 = (1st β€˜π‘…)
rnggrphom.2 𝐽 = (1st β€˜π‘†)
Assertion
Ref Expression
rngogrphom ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ 𝐹 ∈ (𝐺 GrpOpHom 𝐽))

Proof of Theorem rngogrphom
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnggrphom.1 . . 3 𝐺 = (1st β€˜π‘…)
2 eqid 2725 . . 3 ran 𝐺 = ran 𝐺
3 rnggrphom.2 . . 3 𝐽 = (1st β€˜π‘†)
4 eqid 2725 . . 3 ran 𝐽 = ran 𝐽
51, 2, 3, 4rngohomf 37496 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ 𝐹:ran 𝐺⟢ran 𝐽)
61, 2, 3rngohomadd 37499 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) β†’ (πΉβ€˜(π‘₯𝐺𝑦)) = ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)))
76eqcomd 2731 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) β†’ ((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))
87ralrimivva 3191 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))
91rngogrpo 37440 . . . 4 (𝑅 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
103rngogrpo 37440 . . . 4 (𝑆 ∈ RingOps β†’ 𝐽 ∈ GrpOp)
112, 4elghomOLD 37417 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp) β†’ (𝐹 ∈ (𝐺 GrpOpHom 𝐽) ↔ (𝐹:ran 𝐺⟢ran 𝐽 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
129, 10, 11syl2an 594 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (𝐹 ∈ (𝐺 GrpOpHom 𝐽) ↔ (𝐹:ran 𝐺⟢ran 𝐽 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
13123adant3 1129 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ (𝐹 ∈ (𝐺 GrpOpHom 𝐽) ↔ (𝐹:ran 𝐺⟢ran 𝐽 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐽(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
145, 8, 13mpbir2and 711 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ 𝐹 ∈ (𝐺 GrpOpHom 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  ran crn 5673  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7416  1st c1st 7989  GrpOpcgr 30343   GrpOpHom cghomOLD 37413  RingOpscrngo 37424   RingOpsHom crngohom 37490
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-map 8845  df-ablo 30399  df-ghomOLD 37414  df-rngo 37425  df-rngohom 37493
This theorem is referenced by:  rngohom0  37502  rngohomsub  37503  rngokerinj  37505
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