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Theorem grpocl 25602
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 25601 . . 3  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
3 fof 5777 . . 3  |-  ( G : ( X  X.  X ) -onto-> X  ->  G : ( X  X.  X ) --> X )
42, 3syl 17 . 2  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) --> X )
5 fovrn 6425 . 2  |-  ( ( G : ( X  X.  X ) --> X  /\  A  e.  X  /\  B  e.  X
)  ->  ( A G B )  e.  X
)
64, 5syl3an1 1263 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1405    e. wcel 1842    X. cxp 4820   ran crn 4823   -->wf 5564   -onto->wfo 5566  (class class class)co 6277   GrpOpcgr 25588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fo 5574  df-fv 5576  df-ov 6280  df-grpo 25593
This theorem is referenced by:  grpoidinvlem2  25607  grpoidinvlem3  25608  grpo2grp  25636  grpoinvop  25643  grpodivf  25648  grpomuldivass  25651  grpopnpcan2  25655  gxcl  25667  gxcom  25671  ablo4  25689  gxdi  25698  ghgrpOLD  25770  ghsubgolemOLD  25772  rngogcl  25793  vcgcl  25852  nvgcl  25913  ghomgrpilem2  29865  ghomsn  29867  ghomf1olem  29873  ablo4pnp  31604  ghomco  31607  divrngcl  31622  iscringd  31658
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