Theorem ghomsn 10396

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 8132	. . . 4 |- {<.<.A, A>., A>.} e. Grp
4	1, 3	eqeltr 1551	. . 3 |- G e. Grp
5	1	rneqi 3354	. . . . 5 |- ran G = ran {<.<.A, A>., A>.}
6		opex 2796	. . . . . 6 |- <.A, A>. e. V
7	6, 2	rnsnop 3464	. . . . 5 |- ran {<.<.A, A>., A>.} = {A}
8	5, 7	eqtr2 1503	. . . 4 |- {A} = ran G
9	8, 8	elghom 10392	. . 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 701	. 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 3731	. . 3 |- (I |` {A}):{A}-1-1-onto->{A}
12		f1of 3703	. . 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 3738	. . . . . . . 8 |- (x = A -> ((I |` {A})` x) = ((I |` {A})` A))
15	2	snid 2445	. . . . . . . . 9 |- A e. {A}
16		fvresi 3857	. . . . . . . . 9 |- (A e. {A} -> ((I |` {A})` A) = A)
17	15, 16	ax-mp 7	. . . . . . . 8 |- ((I |` {A})` A) = A
18	14, 17	syl6eq 1530	. . . . . . 7 |- (x = A -> ((I |` {A})` x) = A)
19		fveq2 3738	. . . . . . . 8 |- (y = A -> ((I |` {A})` y) = ((I |` {A})` A))
20	19, 17	syl6eq 1530	. . . . . . 7 |- (y = A -> ((I |` {A})` y) = A)
21	18, 20	opreqan12d 3993	. . . . . 6 |- ((x = A /\ y = A) -> (((I |` {A})` x)G((I |` {A})` y)) = (AGA))
22		opreq12 3984	. . . . . 6 |- ((x = A /\ y = A) -> (xGy) = (AGA))
23	21, 22	eqtr4d 1517	. . . . 5 |- ((x = A /\ y = A) -> (((I |` {A})` x)G((I |` {A})` y)) = (xGy))
24		elsn 2431	. . . . 5 |- (x e. {A} <-> x = A)
25		elsn 2431	. . . . 5 |- (y e. {A} <-> y = A)
26	23, 24, 25	syl2anb 458	. . . 4 |- ((x e. {A} /\ y e. {A}) -> (((I |` {A})` x)G((I |` {A})` y)) = (xGy))
27	8	grpcl 8053	. . . . . 6 |- ((G e. Grp /\ x e. {A} /\ y e. {A}) -> (xGy) e. {A})
28	4, 27	mp3an1 907	. . . . 5 |- ((x e. {A} /\ y e. {A}) -> (xGy) e. {A})
29		fvresi 3857	. . . . 5 |- ((xGy) e. {A} -> ((I |` {A})` (xGy)) = (xGy))
30	28, 29	syl 10	. . . 4 |- ((x e. {A} /\ y e. {A}) -> ((I |` {A})` (xGy)) = (xGy))
31	26, 30	eqtr4d 1517	. . 3 |- ((x e. {A} /\ y e. {A}) -> (((I |` {A})` x)G((I |` {A})` y)) = ((I |` {A})` (xGy)))
32	31	rgen2a 1706	. 2 |- A.x e. {A}A.y e. {A} (((I |` {A})` x)G((I |` {A})` y)) = ((I |` {A})` (xGy))
33	10, 13, 32	mpbir2an 734	1 |- (I |` {A}) e. (G GrpHom G)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 960   e. wcel 962  A.wral 1652  Vcvv 1818  {csn 2419  <.cop 2421  Icid 2845  ran crn 3185   |` cres 3186  -->wf 3192  -1-1-onto->wf1o 3195  ` cfv 3196  (class class class)co 3977  Grpcgr 8042   GrpHom cghom 10386

This theorem is referenced by:  ghomgrplem 10397

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-rep 2706  ax-sep 2716  ax-nul 2723  ax-pow 2756  ax-pr 2793  ax-un 2880

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 781  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-ral 1656  df-rex 1657  df-reu 1658  df-v 1819  df-sbc 1949  df-csb 2010  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-pw 2412  df-sn 2422  df-pr 2423  df-op 2426  df-uni 2516  df-br 2633  df-opab 2680  df-id 2849  df-xp 3198  df-rel 3199  df-cnv 3200  df-co 3201  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fun 3206  df-fn 3207  df-f 3208  df-f1 3209  df-fo 3210  df-f1o 3211  df-fv 3212  df-opr 3979  df-oprab 3980  df-grp 8046  df-ghom 10388

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