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Theorem rngoablo2 21859
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 4155 . . 3  |-  ( G
RingOps H  <->  <. G ,  H >.  e.  RingOps )
2 relrngo 21814 . . . . 5  |-  Rel  RingOps
3 brrelex12 4856 . . . . 5  |-  ( ( Rel  RingOps  /\  G RingOps H )  ->  ( G  e. 
_V  /\  H  e.  _V ) )
42, 3mpan 652 . . . 4  |-  ( G
RingOps H  ->  ( G  e.  _V  /\  H  e. 
_V ) )
5 op1stg 6299 . . . 4  |-  ( ( G  e.  _V  /\  H  e.  _V )  ->  ( 1st `  <. G ,  H >. )  =  G )
64, 5syl 16 . . 3  |-  ( G
RingOps H  ->  ( 1st ` 
<. G ,  H >. )  =  G )
71, 6sylbir 205 . 2  |-  ( <. G ,  H >.  e.  RingOps 
->  ( 1st `  <. G ,  H >. )  =  G )
8 eqid 2388 . . 3  |-  ( 1st `  <. G ,  H >. )  =  ( 1st `  <. G ,  H >. )
98rngoablo 21826 . 2  |-  ( <. G ,  H >.  e.  RingOps 
->  ( 1st `  <. G ,  H >. )  e.  AbelOp )
107, 9eqeltrrd 2463 1  |-  ( <. G ,  H >.  e.  RingOps 
->  G  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2900   <.cop 3761   class class class wbr 4154   Rel wrel 4824   ` cfv 5395   1stc1st 6287   AbelOpcablo 21718   RingOpscrngo 21812
This theorem is referenced by:  isdivrngo  21868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-ov 6024  df-1st 6289  df-2nd 6290  df-rngo 21813
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