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Theorem rngoablo2 21089
Description: In a unital ring the addition is an abelian group. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
Assertion
Ref Expression
rngoablo2  |-  ( <. G ,  H >.  e.  RingOps 
->  G  e.  AbelOp )

Proof of Theorem rngoablo2
StepHypRef Expression
1 df-br 4024 . . 3  |-  ( G
RingOps H  <->  <. G ,  H >.  e.  RingOps )
2 relrngo 21044 . . . . 5  |-  Rel  RingOps
3 brrelex12 4726 . . . . 5  |-  ( ( Rel  RingOps  /\  G RingOps H )  ->  ( G  e. 
_V  /\  H  e.  _V ) )
42, 3mpan 651 . . . 4  |-  ( G
RingOps H  ->  ( G  e.  _V  /\  H  e. 
_V ) )
5 op1stg 6132 . . . 4  |-  ( ( G  e.  _V  /\  H  e.  _V )  ->  ( 1st `  <. G ,  H >. )  =  G )
64, 5syl 15 . . 3  |-  ( G
RingOps H  ->  ( 1st ` 
<. G ,  H >. )  =  G )
71, 6sylbir 204 . 2  |-  ( <. G ,  H >.  e.  RingOps 
->  ( 1st `  <. G ,  H >. )  =  G )
8 eqid 2283 . . 3  |-  ( 1st `  <. G ,  H >. )  =  ( 1st `  <. G ,  H >. )
98rngoablo 21056 . 2  |-  ( <. G ,  H >.  e.  RingOps 
->  ( 1st `  <. G ,  H >. )  e.  AbelOp )
107, 9eqeltrrd 2358 1  |-  ( <. G ,  H >.  e.  RingOps 
->  G  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023   Rel wrel 4694   ` cfv 5255   1stc1st 6120   AbelOpcablo 20948   RingOpscrngo 21042
This theorem is referenced by:  isdivrngo  21098  rngoinvcl  25421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-rngo 21043
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