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Theorem grpsubf 15918
Description: Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.)
Hypotheses
Ref Expression
grpsubcl.b  |-  B  =  ( Base `  G
)
grpsubcl.m  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
grpsubf  |-  ( G  e.  Grp  ->  .-  :
( B  X.  B
) --> B )

Proof of Theorem grpsubf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpsubcl.b . . . . . . 7  |-  B  =  ( Base `  G
)
2 eqid 2467 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
31, 2grpinvcl 15896 . . . . . 6  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( ( invg `  G ) `  y
)  e.  B )
433adant2 1015 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( ( invg `  G ) `  y
)  e.  B )
5 eqid 2467 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
61, 5grpcl 15864 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  ( ( invg `  G ) `  y
)  e.  B )  ->  ( x ( +g  `  G ) ( ( invg `  G ) `  y
) )  e.  B
)
74, 6syld3an3 1273 . . . 4  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  G ) ( ( invg `  G
) `  y )
)  e.  B )
873expb 1197 . . 3  |-  ( ( G  e.  Grp  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  G
) ( ( invg `  G ) `
 y ) )  e.  B )
98ralrimivva 2885 . 2  |-  ( G  e.  Grp  ->  A. x  e.  B  A. y  e.  B  ( x
( +g  `  G ) ( ( invg `  G ) `  y
) )  e.  B
)
10 grpsubcl.m . . . 4  |-  .-  =  ( -g `  G )
111, 5, 2, 10grpsubfval 15893 . . 3  |-  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x ( +g  `  G ) ( ( invg `  G
) `  y )
) )
1211fmpt2 6848 . 2  |-  ( A. x  e.  B  A. y  e.  B  (
x ( +g  `  G
) ( ( invg `  G ) `
 y ) )  e.  B  <->  .-  : ( B  X.  B ) --> B )
139, 12sylib 196 1  |-  ( G  e.  Grp  ->  .-  :
( B  X.  B
) --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   A.wral 2814    X. cxp 4997   -->wf 5582   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   Grpcgrp 15723   invgcminusg 15724   -gcsg 15726
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  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-reu 2821  df-rmo 2822  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-pw 4012  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-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-0g 14693  df-mnd 15728  df-grp 15858  df-minusg 15859  df-sbg 15860
This theorem is referenced by:  grpsubcl  15919  cnfldsub  18217  distgp  20333  indistgp  20334  clssubg  20342  tgphaus  20350  divstgplem  20354  nrmmetd  20830  isngp2  20852  isngp3  20853  ngpds  20858  ngptgp  20885  tngnm  20900  tngngp2  20901  rrxds  21560
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