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Theorem grplmulf1o 15984
Description: Left multiplication by a group element is a bijection on any group. (Contributed by Mario Carneiro, 17-Jan-2015.)
Hypotheses
Ref Expression
grplmulf1o.b  |-  B  =  ( Base `  G
)
grplmulf1o.p  |-  .+  =  ( +g  `  G )
grplmulf1o.n  |-  F  =  ( x  e.  B  |->  ( X  .+  x
) )
Assertion
Ref Expression
grplmulf1o  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  F : B -1-1-onto-> B )
Distinct variable groups:    x, B    x, G    x,  .+    x, X
Allowed substitution hint:    F( x)

Proof of Theorem grplmulf1o
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 grplmulf1o.n . 2  |-  F  =  ( x  e.  B  |->  ( X  .+  x
) )
2 grplmulf1o.b . . . 4  |-  B  =  ( Base `  G
)
3 grplmulf1o.p . . . 4  |-  .+  =  ( +g  `  G )
42, 3grpcl 15935 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  x  e.  B )  ->  ( X  .+  x
)  e.  B )
543expa 1196 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  x  e.  B
)  ->  ( X  .+  x )  e.  B
)
6 simpl 457 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  G  e.  Grp )
7 eqid 2467 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
82, 7grpinvcl 15967 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( invg `  G ) `  X
)  e.  B )
96, 8jca 532 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( G  e.  Grp  /\  ( ( invg `  G ) `  X
)  e.  B ) )
102, 3grpcl 15935 . . . 4  |-  ( ( G  e.  Grp  /\  ( ( invg `  G ) `  X
)  e.  B  /\  y  e.  B )  ->  ( ( ( invg `  G ) `
 X )  .+  y )  e.  B
)
11103expa 1196 . . 3  |-  ( ( ( G  e.  Grp  /\  ( ( invg `  G ) `  X
)  e.  B )  /\  y  e.  B
)  ->  ( (
( invg `  G ) `  X
)  .+  y )  e.  B )
129, 11sylan 471 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  y  e.  B
)  ->  ( (
( invg `  G ) `  X
)  .+  y )  e.  B )
13 eqcom 2476 . . 3  |-  ( x  =  ( ( ( invg `  G
) `  X )  .+  y )  <->  ( (
( invg `  G ) `  X
)  .+  y )  =  x )
146adantr 465 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  G  e.  Grp )
1512adantrl 715 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( ( invg `  G ) `
 X )  .+  y )  e.  B
)
16 simprl 755 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  x  e.  B )
17 simplr 754 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  X  e.  B )
182, 3grplcan 15974 . . . . 5  |-  ( ( G  e.  Grp  /\  ( ( ( ( invg `  G
) `  X )  .+  y )  e.  B  /\  x  e.  B  /\  X  e.  B
) )  ->  (
( X  .+  (
( ( invg `  G ) `  X
)  .+  y )
)  =  ( X 
.+  x )  <->  ( (
( invg `  G ) `  X
)  .+  y )  =  x ) )
1914, 15, 16, 17, 18syl13anc 1230 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( X  .+  ( ( ( invg `  G ) `
 X )  .+  y ) )  =  ( X  .+  x
)  <->  ( ( ( invg `  G
) `  X )  .+  y )  =  x ) )
20 eqid 2467 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
212, 3, 20, 7grprinv 15969 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  (
( invg `  G ) `  X
) )  =  ( 0g `  G ) )
2221adantr 465 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( X  .+  (
( invg `  G ) `  X
) )  =  ( 0g `  G ) )
2322oveq1d 6310 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( X  .+  ( ( invg `  G ) `  X
) )  .+  y
)  =  ( ( 0g `  G ) 
.+  y ) )
248adantr 465 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( invg `  G ) `  X
)  e.  B )
25 simprr 756 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
y  e.  B )
262, 3grpass 15936 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  ( ( invg `  G ) `  X
)  e.  B  /\  y  e.  B )
)  ->  ( ( X  .+  ( ( invg `  G ) `
 X ) ) 
.+  y )  =  ( X  .+  (
( ( invg `  G ) `  X
)  .+  y )
) )
2714, 17, 24, 25, 26syl13anc 1230 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( X  .+  ( ( invg `  G ) `  X
) )  .+  y
)  =  ( X 
.+  ( ( ( invg `  G
) `  X )  .+  y ) ) )
282, 3, 20grplid 15952 . . . . . . 7  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( ( 0g `  G )  .+  y
)  =  y )
2928ad2ant2rl 748 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( 0g `  G )  .+  y
)  =  y )
3023, 27, 293eqtr3d 2516 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( X  .+  (
( ( invg `  G ) `  X
)  .+  y )
)  =  y )
3130eqeq1d 2469 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( X  .+  ( ( ( invg `  G ) `
 X )  .+  y ) )  =  ( X  .+  x
)  <->  y  =  ( X  .+  x ) ) )
3219, 31bitr3d 255 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( ( ( invg `  G
) `  X )  .+  y )  =  x  <-> 
y  =  ( X 
.+  x ) ) )
3313, 32syl5bb 257 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x  =  ( ( ( invg `  G ) `  X
)  .+  y )  <->  y  =  ( X  .+  x ) ) )
341, 5, 12, 33f1o2d 6522 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  F : B -1-1-onto-> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4511   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   0gc0g 14712   Grpcgrp 15925   invgcminusg 15926
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930
This theorem is referenced by:  sylow1lem2  16492  sylow2blem1  16513
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