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Theorem grplmulf1o 12949
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 12890 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  x  e.  B )  ->  ( X  .+  x
)  e.  B )
543expa 1203 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  x  e.  B
)  ->  ( X  .+  x )  e.  B
)
6 eqid 2177 . . . 4  |-  ( invg `  G )  =  ( invg `  G )
72, 6grpinvcl 12926 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( invg `  G ) `  X
)  e.  B )
82, 3grpcl 12890 . . . 4  |-  ( ( G  e.  Grp  /\  ( ( invg `  G ) `  X
)  e.  B  /\  y  e.  B )  ->  ( ( ( invg `  G ) `
 X )  .+  y )  e.  B
)
983expa 1203 . . 3  |-  ( ( ( G  e.  Grp  /\  ( ( invg `  G ) `  X
)  e.  B )  /\  y  e.  B
)  ->  ( (
( invg `  G ) `  X
)  .+  y )  e.  B )
107, 9syldanl 449 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  y  e.  B
)  ->  ( (
( invg `  G ) `  X
)  .+  y )  e.  B )
11 eqcom 2179 . . 3  |-  ( x  =  ( ( ( invg `  G
) `  X )  .+  y )  <->  ( (
( invg `  G ) `  X
)  .+  y )  =  x )
12 simpll 527 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  G  e.  Grp )
1310adantrl 478 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( ( invg `  G ) `
 X )  .+  y )  e.  B
)
14 simprl 529 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  x  e.  B )
15 simplr 528 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  X  e.  B )
162, 3grplcan 12937 . . . . 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 ) )
1712, 13, 14, 15, 16syl13anc 1240 . . . 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 ) )
18 eqid 2177 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
192, 3, 18, 6grprinv 12928 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  (
( invg `  G ) `  X
) )  =  ( 0g `  G ) )
2019adantr 276 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( X  .+  (
( invg `  G ) `  X
) )  =  ( 0g `  G ) )
2120oveq1d 5892 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( X  .+  ( ( invg `  G ) `  X
) )  .+  y
)  =  ( ( 0g `  G ) 
.+  y ) )
227adantr 276 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( invg `  G ) `  X
)  e.  B )
23 simprr 531 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
y  e.  B )
242, 3grpass 12891 . . . . . . 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 )
) )
2512, 15, 22, 23, 24syl13anc 1240 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( X  .+  ( ( invg `  G ) `  X
) )  .+  y
)  =  ( X 
.+  ( ( ( invg `  G
) `  X )  .+  y ) ) )
262, 3, 18grplid 12911 . . . . . . 7  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( ( 0g `  G )  .+  y
)  =  y )
2726ad2ant2rl 511 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( 0g `  G )  .+  y
)  =  y )
2821, 25, 273eqtr3d 2218 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( X  .+  (
( ( invg `  G ) `  X
)  .+  y )
)  =  y )
2928eqeq1d 2186 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( X  .+  ( ( ( invg `  G ) `
 X )  .+  y ) )  =  ( X  .+  x
)  <->  y  =  ( X  .+  x ) ) )
3017, 29bitr3d 190 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( ( ( invg `  G
) `  X )  .+  y )  =  x  <-> 
y  =  ( X 
.+  x ) ) )
3111, 30bitrid 192 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x  =  ( ( ( invg `  G ) `  X
)  .+  y )  <->  y  =  ( X  .+  x ) ) )
321, 5, 10, 31f1o2d 6078 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  F : B -1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148    |-> cmpt 4066   -1-1-onto->wf1o 5217   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   0gc0g 12710   Grpcgrp 12882   invgcminusg 12883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886
This theorem is referenced by: (None)
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