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.

Step	Hyp	Ref	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 } )
3	1, 2	ax-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
8	7	snid 4006	. . . . . . . . 9 |- A e. { A }
9		fvresi 6006	. . . . . . . . 9 |- ( A e. { A } -> ( ( _I |` { A } ) ` A ) = A )
10	8, 9	ax-mp 5	. . . . . . . 8 |- ( ( _I |` { A } ) ` A ) = A
11	6, 10	syl6eq 2508	. . . . . . 7 |- ( x = A -> ( ( _I |` { A } ) ` x ) = A )
12		fveq2 5792	. . . . . . . 8 |- ( y = A -> ( ( _I |` { A } ) ` y ) = ( ( _I |` { A } ) ` A ) )
13	12, 10	syl6eq 2508	. . . . . . 7 |- ( y = A -> ( ( _I |` { A } ) ` y ) = A )
14	11, 13	oveqan12d 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 ) )
16	14, 15	eqtr4d 2495	. . . . 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 23847	. . . . . . 7 |- { <. <. A , A >. , A >. } e. GrpOp
20	18, 19	eqeltri 2535	. . . . . 6 |- G e. GrpOp
21	18	rneqi 5167	. . . . . . . 8 |- ran G = ran { <. <. A , A >. , A >. }
22		opex 4657	. . . . . . . . 9 |- <. A , A >. e. _V
23	22	rnsnop 5421	. . . . . . . 8 |- ran { <. <. A , A >. , A >. } = { A }
24	21, 23	eqtr2i 2481	. . . . . . 7 |- { A } = ran G
25	24	grpocl 23832	. . . . . 6 |- ( ( G e. GrpOp /\ x e. { A } /\ y e. { A } ) -> ( x G y ) e. { A } )
26	20, 25	mp3an1 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 ) )
28	26, 27	syl 16	. . . 4 |- ( ( x e. { A } /\ y e. { A } ) -> ( ( _I |` { A } ) ` ( x G y ) ) = ( x G y ) )
29	17, 28	eqtr4d 2495	. . 3 |- ( ( x e. { A } /\ y e. { A } ) -> ( ( ( _I |` { A } ) ` x ) G ( ( _I |` { A } ) ` y ) ) = ( ( _I |` { A } ) ` ( x G y ) ) )
30	29	rgen2a 2893	. 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 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 ) ) ) ) )
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 911	1 |- ( _I |` { A } ) e. ( G GrpOpHom G )

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