MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpocl Structured version   Unicode version

Theorem grpocl 24878
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 24877 . . 3  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
3 fof 5793 . . 3  |-  ( G : ( X  X.  X ) -onto-> X  ->  G : ( X  X.  X ) --> X )
42, 3syl 16 . 2  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) --> X )
5 fovrn 6427 . 2  |-  ( ( G : ( X  X.  X ) --> X  /\  A  e.  X  /\  B  e.  X
)  ->  ( A G B )  e.  X
)
64, 5syl3an1 1261 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 973    = wceq 1379    e. wcel 1767    X. cxp 4997   ran crn 5000   -->wf 5582   -onto->wfo 5584  (class class class)co 6282   GrpOpcgr 24864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-ov 6285  df-grpo 24869
This theorem is referenced by:  grpoidinvlem2  24883  grpoidinvlem3  24884  grpo2grp  24912  grpoinvop  24919  grpodivf  24924  grpomuldivass  24927  grpopnpcan2  24931  gxcl  24943  gxcom  24947  ablo4  24965  gxdi  24974  ghgrp  25046  ghsubgolem  25048  rngogcl  25069  vcgcl  25128  nvgcl  25189  ghomgrpilem2  28501  ghomsn  28503  ghomf1olem  28509  ablo4pnp  29945  ghomco  29948  divrngcl  29963  iscringd  29999
  Copyright terms: Public domain W3C validator