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Theorem grpolid 30545
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 30544 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
43simplld 768 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  ran crn 5690  cfv 6563  (class class class)co 7431  GrpOpcgr 30518  GIdcgi 30519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-riota 7388  df-ov 7434  df-grpo 30522  df-gid 30523
This theorem is referenced by:  grpoid  30549  grpoinvid1  30557  grpoinvid2  30558  grpolcan  30559  grpoinvop  30562  ablonncan  30585  vcm  30605  nv0lid  30665  hhssabloilem  31290  grpoeqdivid  37868  ghomidOLD  37876  rngo0lid  37908  rngolz  37909  rngorz  37910  keridl  38019
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