Theorem ghomsn 13631

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 GrpHom G)

Proof of Theorem ghomsn

Step	Hyp	Ref	Expression
1		ghomsn.2	. . . 4 |- G = {<.<.A, A>., A>.}
2		ghomsn.1	. . . . 5 |- A e. _V
3	2	grpsn 9340	. . . 4 |- {<.<.A, A>., A>.} e. Grp
4	1, 3	eqeltri 1967	. . 3 |- G e. Grp
5	1	rneqi 4187	. . . . 5 |- ran G = ran {<.<.A, A>., A>.}
6		opex 3527	. . . . . 6 |- <.A, A>. e. _V
7	6, 2	rnsnop 4375	. . . . 5 |- ran {<.<.A, A>., A>.} = {A}
8	5, 7	eqtr2i 1909	. . . 4 |- {A} = ran G
9	8, 8	elghom 10195	. . 3 |- ((G e. Grp /\ G e. Grp) -> (( _I |` {A}) e. (G GrpHom G) <-> (( _I |` {A}):{A}-->{A} /\ A.x e. {A}A.y e. {A} ((( _I |` {A})` x)G(( _I |` {A})` y)) = (( _I |` {A})` (xGy)))))
10	4, 4, 9	mp2an 761	. 2 |- (( _I |` {A}) e. (G GrpHom G) <-> (( _I |` {A}):{A}-->{A} /\ A.x e. {A}A.y e. {A} ((( _I |` {A})` x)G(( _I |` {A})` y)) = (( _I |` {A})` (xGy))))
11		f1oi 4671	. . 3 |- ( _I |` {A}):{A}-1-1-onto->{A}
12		f1of 4635	. . 3 |- (( _I |` {A}):{A}-1-1-onto->{A} -> ( _I |` {A}):{A}-->{A})
13	11, 12	ax-mp 7	. 2 |- ( _I |` {A}):{A}-->{A}
14		fveq2 4681	. . . . . . . 8 |- (x = A -> (( _I |` {A})` x) = (( _I |` {A})` A))
15	2	snid 3069	. . . . . . . . 9 |- A e. {A}
16		fvresi 4819	. . . . . . . . 9 |- (A e. {A} -> (( _I |` {A})` A) = A)
17	15, 16	ax-mp 7	. . . . . . . 8 |- (( _I |` {A})` A) = A
18	14, 17	syl6eq 1944	. . . . . . 7 |- (x = A -> (( _I |` {A})` x) = A)
19		fveq2 4681	. . . . . . . 8 |- (y = A -> (( _I |` {A})` y) = (( _I |` {A})` A))
20	19, 17	syl6eq 1944	. . . . . . 7 |- (y = A -> (( _I |` {A})` y) = A)
21	18, 20	opreqan12d 4902	. . . . . 6 |- ((x = A /\ y = A) -> ((( _I |` {A})` x)G(( _I |` {A})` y)) = (AGA))
22		opreq12 4891	. . . . . 6 |- ((x = A /\ y = A) -> (xGy) = (AGA))
23	21, 22	eqtr4d 1928	. . . . 5 |- ((x = A /\ y = A) -> ((( _I |` {A})` x)G(( _I |` {A})` y)) = (xGy))
24		elsn 3058	. . . . 5 |- (x e. {A} <-> x = A)
25		elsn 3058	. . . . 5 |- (y e. {A} <-> y = A)
26	23, 24, 25	syl2anb 504	. . . 4 |- ((x e. {A} /\ y e. {A}) -> ((( _I |` {A})` x)G(( _I |` {A})` y)) = (xGy))
27	8	grpcl 9324	. . . . . 6 |- ((G e. Grp /\ x e. {A} /\ y e. {A}) -> (xGy) e. {A})
28	4, 27	mp3an1 1178	. . . . 5 |- ((x e. {A} /\ y e. {A}) -> (xGy) e. {A})
29		fvresi 4819	. . . . 5 |- ((xGy) e. {A} -> (( _I |` {A})` (xGy)) = (xGy))
30	28, 29	syl 12	. . . 4 |- ((x e. {A} /\ y e. {A}) -> (( _I |` {A})` (xGy)) = (xGy))
31	26, 30	eqtr4d 1928	. . 3 |- ((x e. {A} /\ y e. {A}) -> ((( _I |` {A})` x)G(( _I |` {A})` y)) = (( _I |` {A})` (xGy)))
32	31	rgen2a 2160	. 2 |- A.x e. {A}A.y e. {A} ((( _I |` {A})` x)G(( _I |` {A})` y)) = (( _I |` {A})` (xGy))
33	10, 13, 32	mpbir2an 800	1 |- ( _I |` {A}) e. (G GrpHom G)

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  {csn 3044  <.cop 3046   _I cid 3582  ran crn 3987   |` cres 3988  -->wf 3994  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  Grpcgr 9311   GrpHom cghom 10189

This theorem is referenced by:  ghomgrplem 13632

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-grp 9316  df-ghom 10190

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