Theorem rngoaddneg1 33727

Description: Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
ringnegcl.1	|- G = ( 1st ` R )
ringnegcl.2	|- X = ran G
ringnegcl.3	|- N = ( inv ` G )
ringaddneg.4	|- Z = (GId ` G )

Assertion

Ref	Expression
rngoaddneg1	|- ( ( R e. RingOps /\ A e. X ) -> ( A G ( N ` A ) ) = Z )

Proof of Theorem rngoaddneg1

Step	Hyp	Ref	Expression
1		ringnegcl.1	. . 3 |- G = ( 1st ` R )
2	1	rngogrpo 33709	. 2 |- ( R e. RingOps -> G e. GrpOp )
3		ringnegcl.2	. . 3 |- X = ran G
4		ringaddneg.4	. . 3 |- Z = (GId ` G )
5		ringnegcl.3	. . 3 |- N = ( inv ` G )
6	3, 4, 5	grporinv 27381	. 2 |- ( ( G e. GrpOp /\ A e. X ) -> ( A G ( N ` A ) ) = Z )
7	2, 6	sylan 488	1 |- ( ( R e. RingOps /\ A e. X ) -> ( A G ( N ` A ) ) = Z )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   GrpOpcgr 27343  GIdcgi 27344   invcgn 27345   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-rep 4771  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-rngo 33694

This theorem is referenced by:  rngonegmn1l  33740