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

Theorem rngoaddneg2 37578
Description: Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
ringnegcl.1 𝐺 = (1st𝑅)
ringnegcl.2 𝑋 = ran 𝐺
ringnegcl.3 𝑁 = (inv‘𝐺)
ringaddneg.4 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
rngoaddneg2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑍)

Proof of Theorem rngoaddneg2
StepHypRef Expression
1 ringnegcl.1 . . 3 𝐺 = (1st𝑅)
21rngogrpo 37559 . 2 (𝑅 ∈ RingOps → 𝐺 ∈ 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 30451 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑍)
72, 6sylan 578 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  ran crn 5682  cfv 6553  (class class class)co 7423  1st c1st 8000  GrpOpcgr 30414  GIdcgi 30415  invcgn 30416  RingOpscrngo 37543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7379  df-ov 7426  df-1st 8002  df-2nd 8003  df-grpo 30418  df-gid 30419  df-ginv 30420  df-ablo 30470  df-rngo 37544
This theorem is referenced by:  rngonegmn1r  37591
  Copyright terms: Public domain W3C validator