Theorem ghomsn 27258

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 5671	. . 3 |- ( _I |` { A } ) : { A } -1-1-onto-> { A }
2		f1of 5636	. . 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 3886	. . . . 5 |- ( x e. { A } <-> x = A )
5		elsn 3886	. . . . 5 |- ( y e. { A } <-> y = A )
6		fveq2 5686	. . . . . . . 8 |- ( x = A -> ( ( _I |` { A } ) ` x ) = ( ( _I |` { A } ) ` A ) )
7		ghomsn.1	. . . . . . . . . 10 |- A e. _V
8	7	snid 3900	. . . . . . . . 9 |- A e. { A }
9		fvresi 5899	. . . . . . . . 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 5686	. . . . . . . 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 6105	. . . . . 6 |- ( ( x = A /\ y = A ) -> ( ( ( _I |` { A } ) ` x ) G ( ( _I |` { A } ) ` y ) ) = ( A G A ) )
15		oveq12 6095	. . . . . 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 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 23653	. . . . . . 7 |- { <. <. A , A >. , A >. } e. GrpOp
20	18, 19	eqeltri 2508	. . . . . 6 |- G e. GrpOp
21	18	rneqi 5061	. . . . . . . 8 |- ran G = ran { <. <. A , A >. , A >. }
22		opex 4551	. . . . . . . . 9 |- <. A , A >. e. _V
23	22	rnsnop 5315	. . . . . . . 8 |- ran { <. <. A , A >. , A >. } = { A }
24	21, 23	eqtr2i 2459	. . . . . . 7 |- { A } = ran G
25	24	grpocl 23638	. . . . . 6 |- ( ( G e. GrpOp /\ x e. { A } /\ y e. { A } ) -> ( x G y ) e. { A } )
26	20, 25	mp3an1 1301	. . . . 5 |- ( ( x e. { A } /\ y e. { A } ) -> ( x G y ) e. { A } )
27		fvresi 5899	. . . . 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 2473	. . 3 |- ( ( x e. { A } /\ y e. { A } ) -> ( ( ( _I |` { A } ) ` x ) G ( ( _I |` { A } ) ` y ) ) = ( ( _I |` { A } ) ` ( x G y ) ) )
30	29	rgen2a 2777	. 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 23801	. . 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 1369   e. wcel 1756   A.wral 2710   _Vcvv 2967   {csn 3872   <.cop 3878   _I cid 4626   ran crn 4836   |` cres 4837   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086   GrpOpcgr 23624   GrpOpHom cghom 23795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-grpo 23629  df-ghom 23796

This theorem is referenced by:  ghomgrplem  27259