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Theorem rngoablo2 22002
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 4205 . . 3  |-  ( G
RingOps H  <->  <. G ,  H >.  e.  RingOps )
2 relrngo 21957 . . . . 5  |-  Rel  RingOps
3 brrelex12 4907 . . . . 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 6351 . . . 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 2435 . . 3  |-  ( 1st `  <. G ,  H >. )  =  ( 1st `  <. G ,  H >. )
98rngoablo 21969 . 2  |-  ( <. G ,  H >.  e.  RingOps 
->  ( 1st `  <. G ,  H >. )  e.  AbelOp )
107, 9eqeltrrd 2510 1  |-  ( <. G ,  H >.  e.  RingOps 
->  G  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   class class class wbr 4204   Rel wrel 4875   ` cfv 5446   1stc1st 6339   AbelOpcablo 21861   RingOpscrngo 21955
This theorem is referenced by:  isdivrngo  22011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-1st 6341  df-2nd 6342  df-rngo 21956
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