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Theorem ghomsn 23995
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 5511 . . 3  |-  (  _I  |`  { A } ) : { A } -1-1-onto-> { A }
2 f1of 5472 . . 3  |-  ( (  _I  |`  { A } ) : { A } -1-1-onto-> { A }  ->  (  _I  |`  { A } ) : { A } --> { A }
)
31, 2ax-mp 8 . 2  |-  (  _I  |`  { A } ) : { A } --> { A }
4 elsn 3655 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
5 elsn 3655 . . . . 5  |-  ( y  e.  { A }  <->  y  =  A )
6 fveq2 5525 . . . . . . . 8  |-  ( x  =  A  ->  (
(  _I  |`  { A } ) `  x
)  =  ( (  _I  |`  { A } ) `  A
) )
7 ghomsn.1 . . . . . . . . . 10  |-  A  e. 
_V
87snid 3667 . . . . . . . . 9  |-  A  e. 
{ A }
9 fvresi 5711 . . . . . . . . 9  |-  ( A  e.  { A }  ->  ( (  _I  |`  { A } ) `  A
)  =  A )
108, 9ax-mp 8 . . . . . . . 8  |-  ( (  _I  |`  { A } ) `  A
)  =  A
116, 10syl6eq 2331 . . . . . . 7  |-  ( x  =  A  ->  (
(  _I  |`  { A } ) `  x
)  =  A )
12 fveq2 5525 . . . . . . . 8  |-  ( y  =  A  ->  (
(  _I  |`  { A } ) `  y
)  =  ( (  _I  |`  { A } ) `  A
) )
1312, 10syl6eq 2331 . . . . . . 7  |-  ( y  =  A  ->  (
(  _I  |`  { A } ) `  y
)  =  A )
1411, 13oveqan12d 5877 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A )  ->  ( ( (  _I  |`  { A } ) `
 x ) G ( (  _I  |`  { A } ) `  y
) )  =  ( A G A ) )
15 oveq12 5867 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A )  ->  ( x G y )  =  ( A G A ) )
1614, 15eqtr4d 2318 . . . . 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 20882 . . . . . . 7  |-  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
2018, 19eqeltri 2353 . . . . . 6  |-  G  e. 
GrpOp
2118rneqi 4905 . . . . . . . 8  |-  ran  G  =  ran  { <. <. A ,  A >. ,  A >. }
22 opex 4237 . . . . . . . . 9  |-  <. A ,  A >.  e.  _V
2322rnsnop 5153 . . . . . . . 8  |-  ran  { <. <. A ,  A >. ,  A >. }  =  { A }
2421, 23eqtr2i 2304 . . . . . . 7  |-  { A }  =  ran  G
2524grpocl 20867 . . . . . 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 5711 . . . . 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 2318 . . 3  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
( (  _I  |`  { A } ) `  x
) G ( (  _I  |`  { A } ) `  y
) )  =  ( (  _I  |`  { A } ) `  (
x G y ) ) )
3029rgen2a 2609 . 2  |-  A. x  e.  { A } A. y  e.  { A }  ( ( (  _I  |`  { A } ) `  x
) G ( (  _I  |`  { A } ) `  y
) )  =  ( (  _I  |`  { A } ) `  (
x G y ) )
3124, 24elghom 21030 . . 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   {csn 3640   <.cop 3643    _I cid 4304   ran crn 4690    |` cres 4691   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853   GrpOpHom cghom 21024
This theorem is referenced by:  ghomgrplem  23996
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-grpo 20858  df-ghom 21025
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