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Theorem ghomsn 24010
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 5527 . . 3  |-  (  _I  |`  { A } ) : { A } -1-1-onto-> { A }
2 f1of 5488 . . 3  |-  ( (  _I  |`  { A } ) : { A } -1-1-onto-> { A }  ->  (  _I  |`  { A } ) : { A } --> { A }
)
31, 2ax-mp 8 . 2  |-  (  _I  |`  { A } ) : { A } --> { A }
4 elsn 3668 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
5 elsn 3668 . . . . 5  |-  ( y  e.  { A }  <->  y  =  A )
6 fveq2 5541 . . . . . . . 8  |-  ( x  =  A  ->  (
(  _I  |`  { A } ) `  x
)  =  ( (  _I  |`  { A } ) `  A
) )
7 ghomsn.1 . . . . . . . . . 10  |-  A  e. 
_V
87snid 3680 . . . . . . . . 9  |-  A  e. 
{ A }
9 fvresi 5727 . . . . . . . . 9  |-  ( A  e.  { A }  ->  ( (  _I  |`  { A } ) `  A
)  =  A )
108, 9ax-mp 8 . . . . . . . 8  |-  ( (  _I  |`  { A } ) `  A
)  =  A
116, 10syl6eq 2344 . . . . . . 7  |-  ( x  =  A  ->  (
(  _I  |`  { A } ) `  x
)  =  A )
12 fveq2 5541 . . . . . . . 8  |-  ( y  =  A  ->  (
(  _I  |`  { A } ) `  y
)  =  ( (  _I  |`  { A } ) `  A
) )
1312, 10syl6eq 2344 . . . . . . 7  |-  ( y  =  A  ->  (
(  _I  |`  { A } ) `  y
)  =  A )
1411, 13oveqan12d 5893 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A )  ->  ( ( (  _I  |`  { A } ) `
 x ) G ( (  _I  |`  { A } ) `  y
) )  =  ( A G A ) )
15 oveq12 5883 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A )  ->  ( x G y )  =  ( A G A ) )
1614, 15eqtr4d 2331 . . . . 5  |-  ( ( x  =  A  /\  y  =  A )  ->  ( ( (  _I  |`  { A } ) `
 x ) G ( (  _I  |`  { A } ) `  y
) )  =  ( x G y ) )
174, 5, 16syl2anb 465 . . . 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 20898 . . . . . . 7  |-  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
2018, 19eqeltri 2366 . . . . . 6  |-  G  e. 
GrpOp
2118rneqi 4921 . . . . . . . 8  |-  ran  G  =  ran  { <. <. A ,  A >. ,  A >. }
22 opex 4253 . . . . . . . . 9  |-  <. A ,  A >.  e.  _V
2322rnsnop 5169 . . . . . . . 8  |-  ran  { <. <. A ,  A >. ,  A >. }  =  { A }
2421, 23eqtr2i 2317 . . . . . . 7  |-  { A }  =  ran  G
2524grpocl 20883 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  x  e.  { A }  /\  y  e.  { A } )  ->  (
x G y )  e.  { A }
)
2620, 25mp3an1 1264 . . . . 5  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
x G y )  e.  { A }
)
27 fvresi 5727 . . . . 5  |-  ( ( x G y )  e.  { A }  ->  ( (  _I  |`  { A } ) `  (
x G y ) )  =  ( x G y ) )
2826, 27syl 15 . . . 4  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
(  _I  |`  { A } ) `  (
x G y ) )  =  ( x G y ) )
2917, 28eqtr4d 2331 . . 3  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
( (  _I  |`  { A } ) `  x
) G ( (  _I  |`  { A } ) `  y
) )  =  ( (  _I  |`  { A } ) `  (
x G y ) ) )
3029rgen2a 2622 . 2  |-  A. x  e.  { A } A. y  e.  { A }  ( ( (  _I  |`  { A } ) `  x
) G ( (  _I  |`  { A } ) `  y
) )  =  ( (  _I  |`  { A } ) `  (
x G y ) )
3124, 24elghom 21046 . . 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 653 . 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 886 1  |-  (  _I  |`  { A } )  e.  ( G GrpOpHom  G )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   {csn 3653   <.cop 3656    _I cid 4320   ran crn 4706    |` cres 4707   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869   GrpOpHom cghom 21040
This theorem is referenced by:  ghomgrplem  24011
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-pw 3640  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-oprab 5878  df-mpt2 5879  df-grpo 20874  df-ghom 21041
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