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Theorem rngoablo2 21105
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 4040 . . 3  |-  ( G
RingOps H  <->  <. G ,  H >.  e.  RingOps )
2 relrngo 21060 . . . . 5  |-  Rel  RingOps
3 brrelex12 4742 . . . . 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 6148 . . . 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 2296 . . 3  |-  ( 1st `  <. G ,  H >. )  =  ( 1st `  <. G ,  H >. )
98rngoablo 21072 . 2  |-  ( <. G ,  H >.  e.  RingOps 
->  ( 1st `  <. G ,  H >. )  e.  AbelOp )
107, 9eqeltrrd 2371 1  |-  ( <. G ,  H >.  e.  RingOps 
->  G  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039   Rel wrel 4710   ` cfv 5271   1stc1st 6136   AbelOpcablo 20964   RingOpscrngo 21058
This theorem is referenced by:  isdivrngo  21114  rngoinvcl  25524
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-rngo 21059
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