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Theorem grpodivid 28328
Description: Division of a group member by itself. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1 𝑋 = ran 𝐺
grpdivf.3 𝐷 = ( /𝑔𝐺)
grpdivid.3 𝑈 = (GId‘𝐺)
Assertion
Ref Expression
grpodivid ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 𝑈)

Proof of Theorem grpodivid
StepHypRef Expression
1 grpdivf.1 . . . 4 𝑋 = ran 𝐺
2 eqid 2824 . . . 4 (inv‘𝐺) = (inv‘𝐺)
3 grpdivf.3 . . . 4 𝐷 = ( /𝑔𝐺)
41, 2, 3grpodivval 28321 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐴𝑋) → (𝐴𝐷𝐴) = (𝐴𝐺((inv‘𝐺)‘𝐴)))
543anidm23 1418 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = (𝐴𝐺((inv‘𝐺)‘𝐴)))
6 grpdivid.3 . . 3 𝑈 = (GId‘𝐺)
71, 6, 2grporinv 28313 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺((inv‘𝐺)‘𝐴)) = 𝑈)
85, 7eqtrd 2859 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  ran crn 5543  cfv 6343  (class class class)co 7149  GrpOpcgr 28275  GIdcgi 28276  invcgn 28277   /𝑔 cgs 28278
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-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-grpo 28279  df-gid 28280  df-ginv 28281  df-gdiv 28282
This theorem is referenced by:  ablonncan  28342  grpoeqdivid  35264
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