Theorem ghomsn 10344

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 8088	. . . 4 |- {<.<.A, A>., A>.} e. Grp
4	1, 3	eqeltr 1542	. . 3 |- G e. Grp
5	1	rneqi 3336	. . . . 5 |- ran G = ran {<.<.A, A>., A>.}
6		opex 2778	. . . . . 6 |- <.A, A>. e. V
7	6, 2	rnsnop 3446	. . . . 5 |- ran {<.<.A, A>., A>.} = {A}
8	5, 7	eqtr2 1494	. . . 4 |- {A} = ran G
9	8, 8	elghom 10340	. . 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 696	. 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 3712	. . 3 |- (I |` {A}):{A}-1-1-onto->{A}
12		f1of 3684	. . 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 3719	. . . . . . . 8 |- (x = A -> ((I |` {A})` x) = ((I |` {A})` A))
15	2	snid 2432	. . . . . . . . 9 |- A e. {A}
16		fvresi 3838	. . . . . . . . 9 |- (A e. {A} -> ((I |` {A})` A) = A)
17	15, 16	ax-mp 7	. . . . . . . 8 |- ((I |` {A})` A) = A
18	14, 17	syl6eq 1521	. . . . . . 7 |- (x = A -> ((I |` {A})` x) = A)
19		fveq2 3719	. . . . . . . 8 |- (y = A -> ((I |` {A})` y) = ((I |` {A})` A))
20	19, 17	syl6eq 1521	. . . . . . 7 |- (y = A -> ((I |` {A})` y) = A)
21	18, 20	opreqan12d 3974	. . . . . 6 |- ((x = A /\ y = A) -> (((I |` {A})` x)G((I |` {A})` y)) = (AGA))
22		opreq12 3965	. . . . . 6 |- ((x = A /\ y = A) -> (xGy) = (AGA))
23	21, 22	eqtr4d 1508	. . . . 5 |- ((x = A /\ y = A) -> (((I |` {A})` x)G((I |` {A})` y)) = (xGy))
24		elsn 2418	. . . . 5 |- (x e. {A} <-> x = A)
25		elsn 2418	. . . . 5 |- (y e. {A} <-> y = A)
26	23, 24, 25	syl2anb 455	. . . 4 |- ((x e. {A} /\ y e. {A}) -> (((I |` {A})` x)G((I |` {A})` y)) = (xGy))
27	8	grpcl 8006	. . . . . 6 |- ((G e. Grp /\ x e. {A} /\ y e. {A}) -> (xGy) e. {A})
28	4, 27	mp3an1 902	. . . . 5 |- ((x e. {A} /\ y e. {A}) -> (xGy) e. {A})
29		fvresi 3838	. . . . 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 1508	. . 3 |- ((x e. {A} /\ y e. {A}) -> (((I |` {A})` x)G((I |` {A})` y)) = ((I |` {A})` (xGy)))
32	31	rgen2a 1697	. 2 |- A.x e. {A}A.y e. {A} (((I |` {A})` x)G((I |` {A})` y)) = ((I |` {A})` (xGy))
33	10, 13, 32	mpbir2an 729	1 |- (I |` {A}) e. (G GrpHom G)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  A.wral 1643  Vcvv 1808  {csn 2406  <.cop 2408  Icid 2827  ran crn 3167   |` cres 3168  -->wf 3174  -1-1-onto->wf1o 3177  ` cfv 3178  (class class class)co 3958  Grpcgr 7995   GrpHom cghom 10334

This theorem is referenced by:  ghomgrplem 10345

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-opr 3960  df-oprab 3961  df-grp 7999  df-ghom 10336

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