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Theorem rngogrphom 37352
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 2726 . . 3 ran 𝐺 = ran 𝐺
3 rnggrphom.2 . . 3 𝐽 = (1st β€˜π‘†)
4 eqid 2726 . . 3 ran 𝐽 = ran 𝐽
51, 2, 3, 4rngohomf 37347 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ 𝐹:ran 𝐺⟢ran 𝐽)
61, 2, 3rngohomadd 37350 . . . 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 2732 . . 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 3194 . 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 37291 . . . 4 (𝑅 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
103rngogrpo 37291 . . . 4 (𝑆 ∈ RingOps β†’ 𝐽 ∈ GrpOp)
112, 4elghomOLD 37268 . . . 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 595 . . 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 710 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  ran crn 5670  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405  1st c1st 7972  GrpOpcgr 30251   GrpOpHom cghomOLD 37264  RingOpscrngo 37275   RingOpsHom crngohom 37341
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-map 8824  df-ablo 30307  df-ghomOLD 37265  df-rngo 37276  df-rngohom 37344
This theorem is referenced by:  rngohom0  37353  rngohomsub  37354  rngokerinj  37356
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