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.

Step	Hyp	Ref	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 } )
3	1, 2	ax-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
8	7	snid 4030	. . . . . . . . 9 |- A e. { A }
9		fvresi 6105	. . . . . . . . 9 |- ( A e. { A } -> ( ( _I |` { A } ) ` A ) = A )
10	8, 9	ax-mp 5	. . . . . . . 8 |- ( ( _I |` { A } ) ` A ) = A
11	6, 10	syl6eq 2486	. . . . . . 7 |- ( x = A -> ( ( _I |` { A } ) ` x ) = A )
12		fveq2 5881	. . . . . . . 8 |- ( y = A -> ( ( _I |` { A } ) ` y ) = ( ( _I |` { A } ) ` A ) )
13	12, 10	syl6eq 2486	. . . . . . 7 |- ( y = A -> ( ( _I |` { A } ) ` y ) = A )
14	11, 13	oveqan12d 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 ) )
16	14, 15	eqtr4d 2473	. . . . 5 |- ( ( x = A /\ y = A ) -> ( ( ( _I |` { A } ) ` x ) G ( ( _I |` { A } ) ` y ) ) = ( x G y ) )
17	4, 5, 16	syl2anb 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 >. }
19	7	grposn 25779	. . . . . . 7 |- { <. <. A , A >. , A >. } e. GrpOp
20	18, 19	eqeltri 2513	. . . . . 6 |- G e. GrpOp
21	18	rneqi 5081	. . . . . . . 8 |- ran G = ran { <. <. A , A >. , A >. }
22		opex 4686	. . . . . . . . 9 |- <. A , A >. e. _V
23	22	rnsnop 5337	. . . . . . . 8 |- ran { <. <. A , A >. , A >. } = { A }
24	21, 23	eqtr2i 2459	. . . . . . 7 |- { A } = ran G
25	24	grpocl 25764	. . . . . 6 |- ( ( G e. GrpOp /\ x e. { A } /\ y e. { A } ) -> ( x G y ) e. { A } )
26	20, 25	mp3an1 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 ) )
28	26, 27	syl 17	. . . 4 |- ( ( x e. { A } /\ y e. { A } ) -> ( ( _I |` { A } ) ` ( x G y ) ) = ( x G y ) )
29	17, 28	eqtr4d 2473	. . 3 |- ( ( x e. { A } /\ y e. { A } ) -> ( ( ( _I |` { A } ) ` x ) G ( ( _I |` { A } ) ` y ) ) = ( ( _I |` { A } ) ` ( x G y ) ) )
30	29	rgen2a 2859	. 2 |- A. x e. { A } A. y e. { A } ( ( ( _I |` { A } ) ` x ) G ( ( _I |` { A } ) ` y ) ) = ( ( _I |` { A } ) ` ( x G y ) )
31	24, 24	elghomOLD 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 ) ) ) ) )
32	20, 20, 31	mp2an 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 ) ) ) )
33	3, 30, 32	mpbir2an 928	1 |- ( _I |` { A } ) e. ( G GrpOpHom G )

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