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Theorem ltrinvlem 25509
Description: The converse of a left translation. The term  A is a constant. (Contributed by FL, 30-Apr-2012.)
Hypotheses
Ref Expression
ltrdom.1  |-  F  =  ( x  e.  X  |->  ( A G x ) )
ltrdom.2  |-  X  =  ran  G
ltrinvlem.2  |-  D  =  (  /g  `  G
)
ltrinvlem.3  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
ltrinvlem  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  `' F  =  ( x  e.  X  |->  ( ( N `  A ) G x ) ) )
Distinct variable groups:    x, A    x, G    x, N    x, X
Allowed substitution hints:    D( x)    F( x)

Proof of Theorem ltrinvlem
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ltrdom.1 . . . . . . . 8  |-  F  =  ( x  e.  X  |->  ( A G x ) )
2 ltrdom.2 . . . . . . . 8  |-  X  =  ran  G
31, 2ltrooo 25507 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  F : X -1-1-onto-> X )
43adantr 451 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  F : X
-1-1-onto-> X )
5 simpll 730 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  G  e.  GrpOp
)
6 ltrinvlem.3 . . . . . . . . 9  |-  N  =  ( inv `  G
)
72, 6grpoinvcl 20909 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
87adantr 451 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( N `  A )  e.  X
)
9 simpr 447 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  x  e.  X )
102grpocl 20883 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X  /\  x  e.  X )  ->  (
( N `  A
) G x )  e.  X )
115, 8, 9, 10syl3anc 1182 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( ( N `  A ) G x )  e.  X )
124, 11jca 518 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( F : X -1-1-onto-> X  /\  ( ( N `  A ) G x )  e.  X ) )
13 simplr 731 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  A  e.  X )
142grpocl 20883 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  (
( N `  A
) G x )  e.  X )  -> 
( A G ( ( N `  A
) G x ) )  e.  X )
155, 13, 11, 14syl3anc 1182 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( A G ( ( N `
 A ) G x ) )  e.  X )
16 oveq2 5882 . . . . . . . 8  |-  ( y  =  ( ( N `
 A ) G x )  ->  ( A G y )  =  ( A G ( ( N `  A
) G x ) ) )
17 oveq2 5882 . . . . . . . . . 10  |-  ( x  =  y  ->  ( A G x )  =  ( A G y ) )
1817cbvmptv 4127 . . . . . . . . 9  |-  ( x  e.  X  |->  ( A G x ) )  =  ( y  e.  X  |->  ( A G y ) )
191, 18eqtri 2316 . . . . . . . 8  |-  F  =  ( y  e.  X  |->  ( A G y ) )
2016, 19fvmptg 5616 . . . . . . 7  |-  ( ( ( ( N `  A ) G x )  e.  X  /\  ( A G ( ( N `  A ) G x ) )  e.  X )  -> 
( F `  (
( N `  A
) G x ) )  =  ( A G ( ( N `
 A ) G x ) ) )
2111, 15, 20syl2anc 642 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( F `  ( ( N `  A ) G x ) )  =  ( A G ( ( N `  A ) G x ) ) )
222grpoass 20886 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  ( N `  A )  e.  X  /\  x  e.  X ) )  -> 
( ( A G ( N `  A
) ) G x )  =  ( A G ( ( N `
 A ) G x ) ) )
2322eqcomd 2301 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  ( N `  A )  e.  X  /\  x  e.  X ) )  -> 
( A G ( ( N `  A
) G x ) )  =  ( ( A G ( N `
 A ) ) G x ) )
245, 13, 8, 9, 23syl13anc 1184 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( A G ( ( N `
 A ) G x ) )  =  ( ( A G ( N `  A
) ) G x ) )
25 eqid 2296 . . . . . . . . . 10  |-  (GId `  G )  =  (GId
`  G )
262, 25, 6grporinv 20912 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
2726adantr 451 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
2827oveq1d 5889 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( ( A G ( N `  A ) ) G x )  =  ( (GId `  G ) G x ) )
292, 25grpolid 20902 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  (
(GId `  G ) G x )  =  x )
3029adantlr 695 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( (GId `  G ) G x )  =  x )
3128, 30eqtrd 2328 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( ( A G ( N `  A ) ) G x )  =  x )
3221, 24, 313eqtrd 2332 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( F `  ( ( N `  A ) G x ) )  =  x )
33 f1ocnvfv 5810 . . . . 5  |-  ( ( F : X -1-1-onto-> X  /\  ( ( N `  A ) G x )  e.  X )  ->  ( ( F `
 ( ( N `
 A ) G x ) )  =  x  ->  ( `' F `  x )  =  ( ( N `
 A ) G x ) ) )
3412, 32, 33sylc 56 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( `' F `  x )  =  ( ( N `
 A ) G x ) )
35 eqid 2296 . . . . . 6  |-  ( x  e.  X  |->  ( ( N `  A ) G x ) )  =  ( x  e.  X  |->  ( ( N `
 A ) G x ) )
3635fvmpt2 5624 . . . . 5  |-  ( ( x  e.  X  /\  ( ( N `  A ) G x )  e.  X )  ->  ( ( x  e.  X  |->  ( ( N `  A ) G x ) ) `
 x )  =  ( ( N `  A ) G x ) )
379, 11, 36syl2anc 642 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( (
x  e.  X  |->  ( ( N `  A
) G x ) ) `  x )  =  ( ( N `
 A ) G x ) )
3834, 37eqtr4d 2331 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( `' F `  x )  =  ( ( x  e.  X  |->  ( ( N `  A ) G x ) ) `
 x ) )
3938ralrimiva 2639 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  A. x  e.  X  ( `' F `  x )  =  ( ( x  e.  X  |->  ( ( N `  A ) G x ) ) `
 x ) )
40 f1ocnv 5501 . . . 4  |-  ( F : X -1-1-onto-> X  ->  `' F : X -1-1-onto-> X )
41 f1ofn 5489 . . . 4  |-  ( `' F : X -1-1-onto-> X  ->  `' F  Fn  X
)
423, 40, 413syl 18 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  `' F  Fn  X )
4335, 2ltrooo 25507 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  (
x  e.  X  |->  ( ( N `  A
) G x ) ) : X -1-1-onto-> X )
447, 43syldan 456 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
x  e.  X  |->  ( ( N `  A
) G x ) ) : X -1-1-onto-> X )
45 f1ofn 5489 . . . 4  |-  ( ( x  e.  X  |->  ( ( N `  A
) G x ) ) : X -1-1-onto-> X  -> 
( x  e.  X  |->  ( ( N `  A ) G x ) )  Fn  X
)
4644, 45syl 15 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
x  e.  X  |->  ( ( N `  A
) G x ) )  Fn  X )
47 nfmpt1 4125 . . . . . 6  |-  F/_ x
( x  e.  X  |->  ( A G x ) )
481, 47nfcxfr 2429 . . . . 5  |-  F/_ x F
4948nfcnv 4876 . . . 4  |-  F/_ x `' F
50 nfmpt1 4125 . . . 4  |-  F/_ x
( x  e.  X  |->  ( ( N `  A ) G x ) )
5149, 50eqfnfv2f 5642 . . 3  |-  ( ( `' F  Fn  X  /\  ( x  e.  X  |->  ( ( N `  A ) G x ) )  Fn  X
)  ->  ( `' F  =  ( x  e.  X  |->  ( ( N `  A ) G x ) )  <->  A. x  e.  X  ( `' F `  x )  =  ( ( x  e.  X  |->  ( ( N `  A ) G x ) ) `
 x ) ) )
5242, 46, 51syl2anc 642 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( `' F  =  (
x  e.  X  |->  ( ( N `  A
) G x ) )  <->  A. x  e.  X  ( `' F `  x )  =  ( ( x  e.  X  |->  ( ( N `  A ) G x ) ) `
 x ) ) )
5339, 52mpbird 223 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  `' F  =  ( x  e.  X  |->  ( ( N `  A ) G x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    e. cmpt 4093   `'ccnv 4704   ran crn 4706    Fn wfn 5266   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871    /g cgs 20872
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876
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