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Theorem rngoaddneg1 35087
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 35069 . 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 28231 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑍)
72, 6sylan 580 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  ran crn 5549  cfv 6348  (class class class)co 7145  1st c1st 7676  GrpOpcgr 28193  GIdcgi 28194  invcgn 28195  RingOpscrngo 35053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-1st 7678  df-2nd 7679  df-grpo 28197  df-gid 28198  df-ginv 28199  df-ablo 28249  df-rngo 35054
This theorem is referenced by:  rngonegmn1l  35100
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