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Theorem grpolid 29500
Description: The identity element of a group is a left 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
grpolid ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺𝐴) = 𝐴)

Proof of Theorem grpolid
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 grpoidval.1 . . 3 𝑋 = ran 𝐺
2 grpoidval.2 . . 3 𝑈 = (GId‘𝐺)
31, 2grpoidinv2 29499 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
43simplld 767 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wrex 3070  ran crn 5635  cfv 6497  (class class class)co 7358  GrpOpcgr 29473  GIdcgi 29474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-riota 7314  df-ov 7361  df-grpo 29477  df-gid 29478
This theorem is referenced by:  grpoid  29504  grpoinvid1  29512  grpoinvid2  29513  grpolcan  29514  grpoinvop  29517  ablonncan  29540  vcm  29560  nv0lid  29620  hhssabloilem  30245  grpoeqdivid  36386  ghomidOLD  36394  rngo0lid  36426  rngolz  36427  rngorz  36428  keridl  36537
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