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Theorem grporid 28289
Description: The identity element of a group is a right identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpoidval.1 𝑋 = ran 𝐺
grpoidval.2 𝑈 = (GId‘𝐺)
Assertion
Ref Expression
grporid ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺𝑈) = 𝐴)

Proof of Theorem grporid
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 grpoidval.1 . . 3 𝑋 = ran 𝐺
2 grpoidval.2 . . 3 𝑈 = (GId‘𝐺)
31, 2grpoidinv2 28287 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑥𝑋 ((𝑥𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑥) = 𝑈)))
4 simplr 768 . 2 ((((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑥𝑋 ((𝑥𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑥) = 𝑈)) → (𝐴𝐺𝑈) = 𝐴)
53, 4syl 17 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺𝑈) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wrex 3133  ran crn 5537  cfv 6336  (class class class)co 7138  GrpOpcgr 28261  GIdcgi 28262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fo 6342  df-fv 6344  df-riota 7096  df-ov 7141  df-grpo 28265  df-gid 28266
This theorem is referenced by:  grporcan  28290  grpoinvid1  28300  grpoinvid2  28301  grponpcan  28315  vc0rid  28345  vcm  28348  nv0rid  28407  rngo0rid  35258  rngolz  35260
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