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Theorem ghomsn 30085
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 5866 . . 3  |-  (  _I  |`  { A } ) : { A } -1-1-onto-> { A }
2 f1of 5831 . . 3  |-  ( (  _I  |`  { A } ) : { A } -1-1-onto-> { A }  ->  (  _I  |`  { A } ) : { A } --> { A }
)
31, 2ax-mp 5 . 2  |-  (  _I  |`  { A } ) : { A } --> { A }
4 elsn 4016 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
5 elsn 4016 . . . . 5  |-  ( y  e.  { A }  <->  y  =  A )
6 fveq2 5881 . . . . . . . 8  |-  ( x  =  A  ->  (
(  _I  |`  { A } ) `  x
)  =  ( (  _I  |`  { A } ) `  A
) )
7 ghomsn.1 . . . . . . . . . 10  |-  A  e. 
_V
87snid 4030 . . . . . . . . 9  |-  A  e. 
{ A }
9 fvresi 6105 . . . . . . . . 9  |-  ( A  e.  { A }  ->  ( (  _I  |`  { A } ) `  A
)  =  A )
108, 9ax-mp 5 . . . . . . . 8  |-  ( (  _I  |`  { A } ) `  A
)  =  A
116, 10syl6eq 2486 . . . . . . 7  |-  ( x  =  A  ->  (
(  _I  |`  { A } ) `  x
)  =  A )
12 fveq2 5881 . . . . . . . 8  |-  ( y  =  A  ->  (
(  _I  |`  { A } ) `  y
)  =  ( (  _I  |`  { A } ) `  A
) )
1312, 10syl6eq 2486 . . . . . . 7  |-  ( y  =  A  ->  (
(  _I  |`  { A } ) `  y
)  =  A )
1411, 13oveqan12d 6324 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A )  ->  ( ( (  _I  |`  { A } ) `
 x ) G ( (  _I  |`  { A } ) `  y
) )  =  ( A G A ) )
15 oveq12 6314 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A )  ->  ( x G y )  =  ( A G A ) )
1614, 15eqtr4d 2473 . . . . 5  |-  ( ( x  =  A  /\  y  =  A )  ->  ( ( (  _I  |`  { A } ) `
 x ) G ( (  _I  |`  { A } ) `  y
) )  =  ( x G y ) )
174, 5, 16syl2anb 481 . . . 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 25779 . . . . . . 7  |-  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
2018, 19eqeltri 2513 . . . . . 6  |-  G  e. 
GrpOp
2118rneqi 5081 . . . . . . . 8  |-  ran  G  =  ran  { <. <. A ,  A >. ,  A >. }
22 opex 4686 . . . . . . . . 9  |-  <. A ,  A >.  e.  _V
2322rnsnop 5337 . . . . . . . 8  |-  ran  { <. <. A ,  A >. ,  A >. }  =  { A }
2421, 23eqtr2i 2459 . . . . . . 7  |-  { A }  =  ran  G
2524grpocl 25764 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  x  e.  { A }  /\  y  e.  { A } )  ->  (
x G y )  e.  { A }
)
2620, 25mp3an1 1347 . . . . 5  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
x G y )  e.  { A }
)
27 fvresi 6105 . . . . 5  |-  ( ( x G y )  e.  { A }  ->  ( (  _I  |`  { A } ) `  (
x G y ) )  =  ( x G y ) )
2826, 27syl 17 . . . 4  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
(  _I  |`  { A } ) `  (
x G y ) )  =  ( x G y ) )
2917, 28eqtr4d 2473 . . 3  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
( (  _I  |`  { A } ) `  x
) G ( (  _I  |`  { A } ) `  y
) )  =  ( (  _I  |`  { A } ) `  (
x G y ) ) )
3029rgen2a 2859 . 2  |-  A. x  e.  { A } A. y  e.  { A }  ( ( (  _I  |`  { A } ) `  x
) G ( (  _I  |`  { A } ) `  y
) )  =  ( (  _I  |`  { A } ) `  (
x G y ) )
3124, 24elghomOLD 25927 . . 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 676 . 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 928 1  |-  (  _I  |`  { A } )  e.  ( G GrpOpHom  G )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087   {csn 4002   <.cop 4008    _I cid 4764   ran crn 4855    |` cres 4856   -->wf 5597   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305   GrpOpcgr 25750   GrpOpHom cghomOLD 25921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-grpo 25755  df-ghomOLD 25922
This theorem is referenced by:  ghomgrplem  30086
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