Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trinv Unicode version

Theorem trinv 25395
Description: The converse of a right translation. The term  A is a constant. (Contributed by FL, 21-Jun-2010.)
Hypotheses
Ref Expression
trfun.2  |-  F  =  ( x  e.  X  |->  ( x G A ) )
trinv.1  |-  X  =  ran  G
trinv.2  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
trinv  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  `' F  =  ( x  e.  X  |->  ( x D A ) ) )
Distinct variable groups:    x, A    x, G    x, X
Allowed substitution hints:    D( x)    F( x)

Proof of Theorem trinv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 trfun.2 . . . . . . . 8  |-  F  =  ( x  e.  X  |->  ( x G A ) )
2 trinv.1 . . . . . . . 8  |-  X  =  ran  G
31, 2trooo 25394 . . . . . . 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 trinv.2 . . . . . . . . . 10  |-  D  =  (  /g  `  G
)
62, 5grpodivcl 20914 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  x  e.  X  /\  A  e.  X )  ->  (
x D A )  e.  X )
763exp 1150 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  ( x  e.  X  ->  ( A  e.  X  ->  (
x D A )  e.  X ) ) )
87com23 72 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( x  e.  X  ->  (
x D A )  e.  X ) ) )
98imp31 421 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( x D A )  e.  X
)
104, 9jca 518 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( F : X -1-1-onto-> X  /\  ( x D A )  e.  X ) )
11 ovex 5883 . . . . . . 7  |-  ( ( x D A ) G A )  e. 
_V
12 oveq1 5865 . . . . . . . 8  |-  ( y  =  ( x D A )  ->  (
y G A )  =  ( ( x D A ) G A ) )
13 oveq1 5865 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x G A )  =  ( y G A ) )
1413cbvmptv 4111 . . . . . . . . 9  |-  ( x  e.  X  |->  ( x G A ) )  =  ( y  e.  X  |->  ( y G A ) )
151, 14eqtri 2303 . . . . . . . 8  |-  F  =  ( y  e.  X  |->  ( y G A ) )
1612, 15fvmptg 5600 . . . . . . 7  |-  ( ( ( x D A )  e.  X  /\  ( ( x D A ) G A )  e.  _V )  ->  ( F `  (
x D A ) )  =  ( ( x D A ) G A ) )
179, 11, 16sylancl 643 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( F `  ( x D A ) )  =  ( ( x D A ) G A ) )
182, 5grponpcan 20919 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  x  e.  X  /\  A  e.  X )  ->  (
( x D A ) G A )  =  x )
19183exp 1150 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  ( x  e.  X  ->  ( A  e.  X  ->  (
( x D A ) G A )  =  x ) ) )
2019com23 72 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( x  e.  X  ->  (
( x D A ) G A )  =  x ) ) )
2120imp31 421 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( (
x D A ) G A )  =  x )
2217, 21eqtrd 2315 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( F `  ( x D A ) )  =  x )
23 f1ocnvfv 5794 . . . . 5  |-  ( ( F : X -1-1-onto-> X  /\  ( x D A )  e.  X )  ->  ( ( F `
 ( x D A ) )  =  x  ->  ( `' F `  x )  =  ( x D A ) ) )
2410, 22, 23sylc 56 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( `' F `  x )  =  ( x D A ) )
25 simpr 447 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  x  e.  X )
26 ovex 5883 . . . . 5  |-  ( x D A )  e. 
_V
27 eqid 2283 . . . . . 6  |-  ( x  e.  X  |->  ( x D A ) )  =  ( x  e.  X  |->  ( x D A ) )
2827fvmpt2 5608 . . . . 5  |-  ( ( x  e.  X  /\  ( x D A )  e.  _V )  ->  ( ( x  e.  X  |->  ( x D A ) ) `  x )  =  ( x D A ) )
2925, 26, 28sylancl 643 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( (
x  e.  X  |->  ( x D A ) ) `  x )  =  ( x D A ) )
3024, 29eqtr4d 2318 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  x  e.  X
)  ->  ( `' F `  x )  =  ( ( x  e.  X  |->  ( x D A ) ) `
 x ) )
3130ralrimiva 2626 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  A. x  e.  X  ( `' F `  x )  =  ( ( x  e.  X  |->  ( x D A ) ) `
 x ) )
32 dff1o4 5480 . . . . 5  |-  ( F : X -1-1-onto-> X  <->  ( F  Fn  X  /\  `' F  Fn  X ) )
3332simprbi 450 . . . 4  |-  ( F : X -1-1-onto-> X  ->  `' F  Fn  X )
343, 33syl 15 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  `' F  Fn  X )
3526rgenw 2610 . . . 4  |-  A. x  e.  X  ( x D A )  e.  _V
3627mptfng 5369 . . . . 5  |-  ( A. x  e.  X  (
x D A )  e.  _V  <->  ( x  e.  X  |->  ( x D A ) )  Fn  X )
3736a1i 10 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A. x  e.  X  ( x D A )  e.  _V  <->  ( x  e.  X  |->  ( x D A ) )  Fn  X ) )
3835, 37mpbii 202 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
x  e.  X  |->  ( x D A ) )  Fn  X )
391cnveqi 4856 . . . . . 6  |-  `' F  =  `' ( x  e.  X  |->  ( x G A ) )
4039fneq1i 5338 . . . . 5  |-  ( `' F  Fn  X  <->  `' (
x  e.  X  |->  ( x G A ) )  Fn  X )
41 nfmpt1 4109 . . . . . . 7  |-  F/_ x
( x  e.  X  |->  ( x G A ) )
4241nfcnv 4860 . . . . . 6  |-  F/_ x `' ( x  e.  X  |->  ( x G A ) )
43 nfmpt1 4109 . . . . . 6  |-  F/_ x
( x  e.  X  |->  ( x D A ) )
4442, 43eqfnfv2f 5626 . . . . 5  |-  ( ( `' ( x  e.  X  |->  ( x G A ) )  Fn  X  /\  ( x  e.  X  |->  ( x D A ) )  Fn  X )  -> 
( `' ( x  e.  X  |->  ( x G A ) )  =  ( x  e.  X  |->  ( x D A ) )  <->  A. x  e.  X  ( `' ( x  e.  X  |->  ( x G A ) ) `  x
)  =  ( ( x  e.  X  |->  ( x D A ) ) `  x ) ) )
4540, 44sylanb 458 . . . 4  |-  ( ( `' F  Fn  X  /\  ( x  e.  X  |->  ( x D A ) )  Fn  X
)  ->  ( `' ( x  e.  X  |->  ( x G A ) )  =  ( x  e.  X  |->  ( x D A ) )  <->  A. x  e.  X  ( `' ( x  e.  X  |->  ( x G A ) ) `  x )  =  ( ( x  e.  X  |->  ( x D A ) ) `  x
) ) )
4639eqeq1i 2290 . . . 4  |-  ( `' F  =  ( x  e.  X  |->  ( x D A ) )  <->  `' ( x  e.  X  |->  ( x G A ) )  =  ( x  e.  X  |->  ( x D A ) ) )
4739fveq1i 5526 . . . . . 6  |-  ( `' F `  x )  =  ( `' ( x  e.  X  |->  ( x G A ) ) `  x )
4847eqeq1i 2290 . . . . 5  |-  ( ( `' F `  x )  =  ( ( x  e.  X  |->  ( x D A ) ) `
 x )  <->  ( `' ( x  e.  X  |->  ( x G A ) ) `  x
)  =  ( ( x  e.  X  |->  ( x D A ) ) `  x ) )
4948ralbii 2567 . . . 4  |-  ( A. x  e.  X  ( `' F `  x )  =  ( ( x  e.  X  |->  ( x D A ) ) `
 x )  <->  A. x  e.  X  ( `' ( x  e.  X  |->  ( x G A ) ) `  x
)  =  ( ( x  e.  X  |->  ( x D A ) ) `  x ) )
5045, 46, 493bitr4g 279 . . 3  |-  ( ( `' F  Fn  X  /\  ( x  e.  X  |->  ( x D A ) )  Fn  X
)  ->  ( `' F  =  ( x  e.  X  |->  ( x D A ) )  <->  A. x  e.  X  ( `' F `  x )  =  ( ( x  e.  X  |->  ( x D A ) ) `
 x ) ) )
5134, 38, 50syl2anc 642 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( `' F  =  (
x  e.  X  |->  ( x D A ) )  <->  A. x  e.  X  ( `' F `  x )  =  ( ( x  e.  X  |->  ( x D A ) ) `
 x ) ) )
5231, 51mpbird 223 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  `' F  =  ( x  e.  X  |->  ( x D A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    e. cmpt 4077   `'ccnv 4688   ran crn 4690    Fn wfn 5250   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853    /g cgs 20856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861
  Copyright terms: Public domain W3C validator