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Theorem grpocl 20867
Description: Closure law for a group operation. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1  |-  X  =  ran  G
Assertion
Ref Expression
grpocl  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )

Proof of Theorem grpocl
StepHypRef Expression
1 grpfo.1 . . . 4  |-  X  =  ran  G
21grpofo 20866 . . 3  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
3 fof 5451 . . 3  |-  ( G : ( X  X.  X ) -onto-> X  ->  G : ( X  X.  X ) --> X )
42, 3syl 15 . 2  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) --> X )
5 fovrn 5990 . 2  |-  ( ( G : ( X  X.  X ) --> X  /\  A  e.  X  /\  B  e.  X
)  ->  ( A G B )  e.  X
)
64, 5syl3an1 1215 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684    X. cxp 4687   ran crn 4690   -->wf 5251   -onto->wfo 5253  (class class class)co 5858   GrpOpcgr 20853
This theorem is referenced by:  grpoidinvlem2  20872  grpoidinvlem3  20873  grpo2grp  20901  grpoinvop  20908  grpodivf  20913  grpomuldivass  20916  grpopnpcan2  20920  gxcl  20932  gxcom  20936  ablo4  20954  gxdi  20963  ghgrp  21035  ghsubgolem  21037  rngogcl  21058  vcgcl  21115  nvgcl  21176  ghomgrpilem2  23993  ghomsn  23995  ghomf1olem  24001  fprodneg  25378  fprodsub  25379  trran2  25393  cmprtr  25396  ltrran2  25403  ltrooo  25404  ltrinvlem  25406  cmpltr2  25407  cmperltr  25409  cmprltr  25410  rltrran  25414  rltrooo  25415  sum2vv  25462  ablo4pnp  26570  ghomco  26573  divrngcl  26588  iscringd  26624
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-grpo 20858
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