Theorem rngoablo2 33708

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

Step	Hyp	Ref	Expression
1		df-br 4654	. . 3 |- ( G RingOps H <-> <. G , H >. e. RingOps )
2		relrngo 33695	. . . . 5 |- Rel RingOps
3		brrelex12 5155	. . . . 5 |- ( ( Rel RingOps /\ G RingOps H ) -> ( G e. _V /\ H e. _V ) )
4	2, 3	mpan 706	. . . 4 |- ( G RingOps H -> ( G e. _V /\ H e. _V ) )
5		op1stg 7180	. . . 4 |- ( ( G e. _V /\ H e. _V ) -> ( 1st ` <. G , H >. ) = G )
6	4, 5	syl 17	. . 3 |- ( G RingOps H -> ( 1st ` <. G , H >. ) = G )
7	1, 6	sylbir 225	. 2 |- ( <. G , H >. e. RingOps -> ( 1st ` <. G , H >. ) = G )
8		eqid 2622	. . 3 |- ( 1st ` <. G , H >. ) = ( 1st ` <. G , H >. )
9	8	rngoablo 33707	. 2 |- ( <. G , H >. e. RingOps -> ( 1st ` <. G , H >. ) e. AbelOp )
10	7, 9	eqeltrrd 2702	1 |- ( <. G , H >. e. RingOps -> G e. AbelOp )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   Rel wrel 5119   ` cfv 5888   1stc1st 7166   AbelOpcablo 27398   RingOpscrngo 33693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-1st 7168  df-2nd 7169  df-rngo 33694

This theorem is referenced by:  isdivrngo  33749