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Theorem rngoaddneg1 34645
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
rngoaddneg1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑍)

Proof of Theorem rngoaddneg1
StepHypRef Expression
1 ringnegcl.1 . . 3 𝐺 = (1st𝑅)
21rngogrpo 34627 . 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, 5grporinv 28081 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑍)
72, 6sylan 572 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  ran crn 5408  cfv 6188  (class class class)co 6976  1st c1st 7499  GrpOpcgr 28043  GIdcgi 28044  invcgn 28045  RingOpscrngo 34611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-1st 7501  df-2nd 7502  df-grpo 28047  df-gid 28048  df-ginv 28049  df-ablo 28099  df-rngo 34612
This theorem is referenced by:  rngonegmn1l  34658
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