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Theorem rngoaddneg2 37458
Description: Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
ringnegcl.1 𝐺 = (1st β€˜π‘…)
ringnegcl.2 𝑋 = ran 𝐺
ringnegcl.3 𝑁 = (invβ€˜πΊ)
ringaddneg.4 𝑍 = (GIdβ€˜πΊ)
Assertion
Ref Expression
rngoaddneg2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺𝐴) = 𝑍)

Proof of Theorem rngoaddneg2
StepHypRef Expression
1 ringnegcl.1 . . 3 𝐺 = (1st β€˜π‘…)
21rngogrpo 37439 . 2 (𝑅 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
3 ringnegcl.2 . . 3 𝑋 = ran 𝐺
4 ringaddneg.4 . . 3 𝑍 = (GIdβ€˜πΊ)
5 ringnegcl.3 . . 3 𝑁 = (invβ€˜πΊ)
63, 4, 5grpolinv 30378 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺𝐴) = 𝑍)
72, 6sylan 578 1 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺𝐴) = 𝑍)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ran crn 5673  β€˜cfv 6542  (class class class)co 7415  1st c1st 7987  GrpOpcgr 30341  GIdcgi 30342  invcgn 30343  RingOpscrngo 37423
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 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737
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 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-1st 7989  df-2nd 7990  df-grpo 30345  df-gid 30346  df-ginv 30347  df-ablo 30397  df-rngo 37424
This theorem is referenced by:  rngonegmn1r  37471
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