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Theorem grpodivcl 21788
Description: Closure of group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1  |-  X  =  ran  G
grpdivf.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
grpodivcl  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  X )

Proof of Theorem grpodivcl
StepHypRef Expression
1 grpdivf.1 . . 3  |-  X  =  ran  G
2 grpdivf.3 . . 3  |-  D  =  (  /g  `  G
)
31, 2grpodivf 21787 . 2  |-  ( G  e.  GrpOp  ->  D :
( X  X.  X
) --> X )
4 fovrn 6175 . 2  |-  ( ( D : ( X  X.  X ) --> X  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  X
)
53, 4syl3an1 1217 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1721    X. cxp 4835   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040   GrpOpcgr 21727    /g cgs 21730
This theorem is referenced by:  grpodivdiv  21789  grponpncan  21796  grpodiveq  21797  ablomuldiv  21830  ablodivdiv4  21832  ablonnncan  21834  ablonnncan1  21836  ghgrp  21909  ablo4pnp  26445  ghomdiv  26449  grpokerinj  26450  dmncan1  26576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-grpo 21732  df-gid 21733  df-ginv 21734  df-gdiv 21735
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