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Theorem grpocl 20920
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 20919 . . 3  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
3 fof 5489 . . 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 6032 . 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 1633    e. wcel 1701    X. cxp 4724   ran crn 4727   -->wf 5288   -onto->wfo 5290  (class class class)co 5900   GrpOpcgr 20906
This theorem is referenced by:  grpoidinvlem2  20925  grpoidinvlem3  20926  grpo2grp  20954  grpoinvop  20961  grpodivf  20966  grpomuldivass  20969  grpopnpcan2  20973  gxcl  20985  gxcom  20989  ablo4  21007  gxdi  21016  ghgrp  21088  ghsubgolem  21090  rngogcl  21111  vcgcl  21170  nvgcl  21231  ghomgrpilem2  24277  ghomsn  24279  ghomf1olem  24285  ablo4pnp  25718  ghomco  25721  divrngcl  25736  iscringd  25772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fo 5298  df-fv 5300  df-ov 5903  df-grpo 20911
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