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Theorem grpodivid 27736
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 2771 . . . 4 (inv‘𝐺) = (inv‘𝐺)
3 grpdivf.3 . . . 4 𝐷 = ( /𝑔𝐺)
41, 2, 3grpodivval 27729 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐴𝑋) → (𝐴𝐷𝐴) = (𝐴𝐺((inv‘𝐺)‘𝐴)))
543anidm23 1531 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = (𝐴𝐺((inv‘𝐺)‘𝐴)))
6 grpdivid.3 . . 3 𝑈 = (GId‘𝐺)
71, 6, 2grporinv 27721 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺((inv‘𝐺)‘𝐴)) = 𝑈)
85, 7eqtrd 2805 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  ran crn 5251  cfv 6030  (class class class)co 6796  GrpOpcgr 27683  GIdcgi 27684  invcgn 27685   /𝑔 cgs 27686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-grpo 27687  df-gid 27688  df-ginv 27689  df-gdiv 27690
This theorem is referenced by:  ablonncan  27751  grpoeqdivid  34012
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