Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngoaddneg2 Unicode version

Theorem rngoaddneg2 26578
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
rngoaddneg2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  A
) G A )  =  Z )

Proof of Theorem rngoaddneg2
StepHypRef Expression
1 ringnegcl.1 . . 3  |-  G  =  ( 1st `  R
)
21rngogrpo 21057 . 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
)
63, 4, 5grpolinv 20895 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G A )  =  Z )
72, 6sylan 457 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( N `  A
) G A )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855   RingOpscrngo 21042
This theorem is referenced by:  rngonegmn1r  26581
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-rngo 21043
  Copyright terms: Public domain W3C validator