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Theorem ghomsn 27444
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 5777 . . 3  |-  (  _I  |`  { A } ) : { A } -1-1-onto-> { A }
2 f1of 5742 . . 3  |-  ( (  _I  |`  { A } ) : { A } -1-1-onto-> { A }  ->  (  _I  |`  { A } ) : { A } --> { A }
)
31, 2ax-mp 5 . 2  |-  (  _I  |`  { A } ) : { A } --> { A }
4 elsn 3992 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
5 elsn 3992 . . . . 5  |-  ( y  e.  { A }  <->  y  =  A )
6 fveq2 5792 . . . . . . . 8  |-  ( x  =  A  ->  (
(  _I  |`  { A } ) `  x
)  =  ( (  _I  |`  { A } ) `  A
) )
7 ghomsn.1 . . . . . . . . . 10  |-  A  e. 
_V
87snid 4006 . . . . . . . . 9  |-  A  e. 
{ A }
9 fvresi 6006 . . . . . . . . 9  |-  ( A  e.  { A }  ->  ( (  _I  |`  { A } ) `  A
)  =  A )
108, 9ax-mp 5 . . . . . . . 8  |-  ( (  _I  |`  { A } ) `  A
)  =  A
116, 10syl6eq 2508 . . . . . . 7  |-  ( x  =  A  ->  (
(  _I  |`  { A } ) `  x
)  =  A )
12 fveq2 5792 . . . . . . . 8  |-  ( y  =  A  ->  (
(  _I  |`  { A } ) `  y
)  =  ( (  _I  |`  { A } ) `  A
) )
1312, 10syl6eq 2508 . . . . . . 7  |-  ( y  =  A  ->  (
(  _I  |`  { A } ) `  y
)  =  A )
1411, 13oveqan12d 6212 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A )  ->  ( ( (  _I  |`  { A } ) `
 x ) G ( (  _I  |`  { A } ) `  y
) )  =  ( A G A ) )
15 oveq12 6202 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A )  ->  ( x G y )  =  ( A G A ) )
1614, 15eqtr4d 2495 . . . . 5  |-  ( ( x  =  A  /\  y  =  A )  ->  ( ( (  _I  |`  { A } ) `
 x ) G ( (  _I  |`  { A } ) `  y
) )  =  ( x G y ) )
174, 5, 16syl2anb 479 . . . 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 23847 . . . . . . 7  |-  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
2018, 19eqeltri 2535 . . . . . 6  |-  G  e. 
GrpOp
2118rneqi 5167 . . . . . . . 8  |-  ran  G  =  ran  { <. <. A ,  A >. ,  A >. }
22 opex 4657 . . . . . . . . 9  |-  <. A ,  A >.  e.  _V
2322rnsnop 5421 . . . . . . . 8  |-  ran  { <. <. A ,  A >. ,  A >. }  =  { A }
2421, 23eqtr2i 2481 . . . . . . 7  |-  { A }  =  ran  G
2524grpocl 23832 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  x  e.  { A }  /\  y  e.  { A } )  ->  (
x G y )  e.  { A }
)
2620, 25mp3an1 1302 . . . . 5  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
x G y )  e.  { A }
)
27 fvresi 6006 . . . . 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 2495 . . 3  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
( (  _I  |`  { A } ) `  x
) G ( (  _I  |`  { A } ) `  y
) )  =  ( (  _I  |`  { A } ) `  (
x G y ) ) )
3029rgen2a 2893 . 2  |-  A. x  e.  { A } A. y  e.  { A }  ( ( (  _I  |`  { A } ) `  x
) G ( (  _I  |`  { A } ) `  y
) )  =  ( (  _I  |`  { A } ) `  (
x G y ) )
3124, 24elghom 23995 . . 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 672 . 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 911 1  |-  (  _I  |`  { A } )  e.  ( G GrpOpHom  G )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   _Vcvv 3071   {csn 3978   <.cop 3984    _I cid 4732   ran crn 4942    |` cres 4943   -->wf 5515   -1-1-onto->wf1o 5518   ` cfv 5519  (class class class)co 6193   GrpOpcgr 23818   GrpOpHom cghom 23989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-grpo 23823  df-ghom 23990
This theorem is referenced by:  ghomgrplem  27445
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