Theorem ghomsn 28503

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 5849	. . 3 |- ( _I |` { A } ) : { A } -1-1-onto-> { A }
2		f1of 5814	. . 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 4041	. . . . 5 |- ( x e. { A } <-> x = A )
5		elsn 4041	. . . . 5 |- ( y e. { A } <-> y = A )
6		fveq2 5864	. . . . . . . 8 |- ( x = A -> ( ( _I |` { A } ) ` x ) = ( ( _I |` { A } ) ` A ) )
7		ghomsn.1	. . . . . . . . . 10 |- A e. _V
8	7	snid 4055	. . . . . . . . 9 |- A e. { A }
9		fvresi 6085	. . . . . . . . 9 |- ( A e. { A } -> ( ( _I |` { A } ) ` A ) = A )
10	8, 9	ax-mp 5	. . . . . . . 8 |- ( ( _I |` { A } ) ` A ) = A
11	6, 10	syl6eq 2524	. . . . . . 7 |- ( x = A -> ( ( _I |` { A } ) ` x ) = A )
12		fveq2 5864	. . . . . . . 8 |- ( y = A -> ( ( _I |` { A } ) ` y ) = ( ( _I |` { A } ) ` A ) )
13	12, 10	syl6eq 2524	. . . . . . 7 |- ( y = A -> ( ( _I |` { A } ) ` y ) = A )
14	11, 13	oveqan12d 6301	. . . . . 6 |- ( ( x = A /\ y = A ) -> ( ( ( _I |` { A } ) ` x ) G ( ( _I |` { A } ) ` y ) ) = ( A G A ) )
15		oveq12 6291	. . . . . 6 |- ( ( x = A /\ y = A ) -> ( x G y ) = ( A G A ) )
16	14, 15	eqtr4d 2511	. . . . 5 |- ( ( x = A /\ y = A ) -> ( ( ( _I |` { A } ) ` x ) G ( ( _I |` { A } ) ` y ) ) = ( x G y ) )
17	4, 5, 16	syl2anb 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 >. }
19	7	grposn 24893	. . . . . . 7 |- { <. <. A , A >. , A >. } e. GrpOp
20	18, 19	eqeltri 2551	. . . . . 6 |- G e. GrpOp
21	18	rneqi 5227	. . . . . . . 8 |- ran G = ran { <. <. A , A >. , A >. }
22		opex 4711	. . . . . . . . 9 |- <. A , A >. e. _V
23	22	rnsnop 5487	. . . . . . . 8 |- ran { <. <. A , A >. , A >. } = { A }
24	21, 23	eqtr2i 2497	. . . . . . 7 |- { A } = ran G
25	24	grpocl 24878	. . . . . 6 |- ( ( G e. GrpOp /\ x e. { A } /\ y e. { A } ) -> ( x G y ) e. { A } )
26	20, 25	mp3an1 1311	. . . . 5 |- ( ( x e. { A } /\ y e. { A } ) -> ( x G y ) e. { A } )
27		fvresi 6085	. . . . 5 |- ( ( x G y ) e. { A } -> ( ( _I |` { A } ) ` ( x G y ) ) = ( x G y ) )
28	26, 27	syl 16	. . . 4 |- ( ( x e. { A } /\ y e. { A } ) -> ( ( _I |` { A } ) ` ( x G y ) ) = ( x G y ) )
29	17, 28	eqtr4d 2511	. . 3 |- ( ( x e. { A } /\ y e. { A } ) -> ( ( ( _I |` { A } ) ` x ) G ( ( _I |` { A } ) ` y ) ) = ( ( _I |` { A } ) ` ( x G y ) ) )
30	29	rgen2a 2891	. 2 |- A. x e. { A } A. y e. { A } ( ( ( _I |` { A } ) ` x ) G ( ( _I |` { A } ) ` y ) ) = ( ( _I |` { A } ) ` ( x G y ) )
31	24, 24	elghom 25041	. . 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 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 ) ) ) )
33	3, 30, 32	mpbir2an 918	1 |- ( _I |` { A } ) e. ( G GrpOpHom G )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2814   _Vcvv 3113   {csn 4027   <.cop 4033   _I cid 4790   ran crn 5000   |` cres 5001   -->wf 5582   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   GrpOpcgr 24864   GrpOpHom cghom 25035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-grpo 24869  df-ghom 25036

This theorem is referenced by:  ghomgrplem  28504