Users' Mathboxes Mathbox for Paul Chapman < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ghomsn Structured version   Unicode version

Theorem ghomsn 25099
Description: The endomorphism of the trivial group. (Contributed by Paul Chapman, 25-Feb-2008.)
Hypotheses
Ref Expression
ghomsn.1  |-  A  e. 
_V
ghomsn.2  |-  G  =  { <. <. A ,  A >. ,  A >. }
Assertion
Ref Expression
ghomsn  |-  (  _I  |`  { A } )  e.  ( G GrpOpHom  G )

Proof of Theorem ghomsn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oi 5713 . . 3  |-  (  _I  |`  { A } ) : { A } -1-1-onto-> { A }
2 f1of 5674 . . 3  |-  ( (  _I  |`  { A } ) : { A } -1-1-onto-> { A }  ->  (  _I  |`  { A } ) : { A } --> { A }
)
31, 2ax-mp 8 . 2  |-  (  _I  |`  { A } ) : { A } --> { A }
4 elsn 3829 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
5 elsn 3829 . . . . 5  |-  ( y  e.  { A }  <->  y  =  A )
6 fveq2 5728 . . . . . . . 8  |-  ( x  =  A  ->  (
(  _I  |`  { A } ) `  x
)  =  ( (  _I  |`  { A } ) `  A
) )
7 ghomsn.1 . . . . . . . . . 10  |-  A  e. 
_V
87snid 3841 . . . . . . . . 9  |-  A  e. 
{ A }
9 fvresi 5924 . . . . . . . . 9  |-  ( A  e.  { A }  ->  ( (  _I  |`  { A } ) `  A
)  =  A )
108, 9ax-mp 8 . . . . . . . 8  |-  ( (  _I  |`  { A } ) `  A
)  =  A
116, 10syl6eq 2484 . . . . . . 7  |-  ( x  =  A  ->  (
(  _I  |`  { A } ) `  x
)  =  A )
12 fveq2 5728 . . . . . . . 8  |-  ( y  =  A  ->  (
(  _I  |`  { A } ) `  y
)  =  ( (  _I  |`  { A } ) `  A
) )
1312, 10syl6eq 2484 . . . . . . 7  |-  ( y  =  A  ->  (
(  _I  |`  { A } ) `  y
)  =  A )
1411, 13oveqan12d 6100 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A )  ->  ( ( (  _I  |`  { A } ) `
 x ) G ( (  _I  |`  { A } ) `  y
) )  =  ( A G A ) )
15 oveq12 6090 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A )  ->  ( x G y )  =  ( A G A ) )
1614, 15eqtr4d 2471 . . . . 5  |-  ( ( x  =  A  /\  y  =  A )  ->  ( ( (  _I  |`  { A } ) `
 x ) G ( (  _I  |`  { A } ) `  y
) )  =  ( x G y ) )
174, 5, 16syl2anb 466 . . . 4  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
( (  _I  |`  { A } ) `  x
) G ( (  _I  |`  { A } ) `  y
) )  =  ( x G y ) )
18 ghomsn.2 . . . . . . 7  |-  G  =  { <. <. A ,  A >. ,  A >. }
197grposn 21803 . . . . . . 7  |-  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
2018, 19eqeltri 2506 . . . . . 6  |-  G  e. 
GrpOp
2118rneqi 5096 . . . . . . . 8  |-  ran  G  =  ran  { <. <. A ,  A >. ,  A >. }
22 opex 4427 . . . . . . . . 9  |-  <. A ,  A >.  e.  _V
2322rnsnop 5350 . . . . . . . 8  |-  ran  { <. <. A ,  A >. ,  A >. }  =  { A }
2421, 23eqtr2i 2457 . . . . . . 7  |-  { A }  =  ran  G
2524grpocl 21788 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  x  e.  { A }  /\  y  e.  { A } )  ->  (
x G y )  e.  { A }
)
2620, 25mp3an1 1266 . . . . 5  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
x G y )  e.  { A }
)
27 fvresi 5924 . . . . 5  |-  ( ( x G y )  e.  { A }  ->  ( (  _I  |`  { A } ) `  (
x G y ) )  =  ( x G y ) )
2826, 27syl 16 . . . 4  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
(  _I  |`  { A } ) `  (
x G y ) )  =  ( x G y ) )
2917, 28eqtr4d 2471 . . 3  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
( (  _I  |`  { A } ) `  x
) G ( (  _I  |`  { A } ) `  y
) )  =  ( (  _I  |`  { A } ) `  (
x G y ) ) )
3029rgen2a 2772 . 2  |-  A. x  e.  { A } A. y  e.  { A }  ( ( (  _I  |`  { A } ) `  x
) G ( (  _I  |`  { A } ) `  y
) )  =  ( (  _I  |`  { A } ) `  (
x G y ) )
3124, 24elghom 21951 . . 3  |-  ( ( G  e.  GrpOp  /\  G  e.  GrpOp )  ->  (
(  _I  |`  { A } )  e.  ( G GrpOpHom  G )  <->  ( (  _I  |`  { A }
) : { A }
--> { A }  /\  A. x  e.  { A } A. y  e.  { A }  ( (
(  _I  |`  { A } ) `  x
) G ( (  _I  |`  { A } ) `  y
) )  =  ( (  _I  |`  { A } ) `  (
x G y ) ) ) ) )
3220, 20, 31mp2an 654 . 2  |-  ( (  _I  |`  { A } )  e.  ( G GrpOpHom  G )  <->  ( (  _I  |`  { A }
) : { A }
--> { A }  /\  A. x  e.  { A } A. y  e.  { A }  ( (
(  _I  |`  { A } ) `  x
) G ( (  _I  |`  { A } ) `  y
) )  =  ( (  _I  |`  { A } ) `  (
x G y ) ) ) )
333, 30, 32mpbir2an 887 1  |-  (  _I  |`  { A } )  e.  ( G GrpOpHom  G )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956   {csn 3814   <.cop 3817    _I cid 4493   ran crn 4879    |` cres 4880   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   GrpOpcgr 21774   GrpOpHom cghom 21945
This theorem is referenced by:  ghomgrplem  25100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-grpo 21779  df-ghom 21946
  Copyright terms: Public domain W3C validator