Theorem subgid 9429

Description: The identity element of a subgroup is the same as its parent's. (Contributed by Paul Chapman, 3-Mar-2008.)

Hypotheses

Ref	Expression
subgid.1	|- U = (Id` G)
subgid.2	|- T = (Id` H)

Assertion

Ref	Expression
subgid	|- (H e. (SubGrp` G) -> T = U)

Proof of Theorem subgid

Step	Hyp	Ref	Expression
1		issubg 9425	. . . . . 6 |- (H e. (SubGrp` G) <-> (G e. Grp /\ H e. Grp /\ H C_ G))
2	1	simp2bi 892	. . . . 5 |- (H e. (SubGrp` G) -> H e. Grp)
3		eqid 1884	. . . . . 6 |- ran H = ran H
4		subgid.2	. . . . . 6 |- T = (Id` H)
5	3, 4	grpidcl 9343	. . . . 5 |- (H e. Grp -> T e. ran H)
6	2, 5	syl 12	. . . 4 |- (H e. (SubGrp` G) -> T e. ran H)
7	3	subgopr 9427	. . . 4 |- (H e. (SubGrp` G) -> ((T e. ran H /\ T e. ran H) -> (THT) = (TGT)))
8	6, 6, 7	mp2and 767	. . 3 |- (H e. (SubGrp` G) -> (THT) = (TGT))
9	3, 4	grplid 9345	. . . 4 |- ((H e. Grp /\ T e. ran H) -> (THT) = T)
10	2, 6, 9	syl11anc 524	. . 3 |- (H e. (SubGrp` G) -> (THT) = T)
11	8, 10	eqtr3d 1927	. 2 |- (H e. (SubGrp` G) -> (TGT) = T)
12	1	simp1bi 891	. . 3 |- (H e. (SubGrp` G) -> G e. Grp)
13		eqid 1884	. . . . 5 |- ran G = ran G
14	13, 3	subgrnss 9428	. . . 4 |- (H e. (SubGrp` G) -> ran H C_ ran G)
15	14, 6	sseldd 2620	. . 3 |- (H e. (SubGrp` G) -> T e. ran G)
16		subgid.1	. . . 4 |- U = (Id` G)
17	13, 16	grpid 9349	. . 3 |- ((G e. Grp /\ T e. ran G) -> (T = U <-> (TGT) = T))
18	12, 15, 17	syl11anc 524	. 2 |- (H e. (SubGrp` G) -> (T = U <-> (TGT) = T))
19	11, 18	mpbird 213	1 |- (H e. (SubGrp` G) -> T = U)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300   C_ wss 2593  ran crn 3987  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312  SubGrpcsubg 9423

This theorem is referenced by:  ssga 9455  cayleylem3 13643

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-rab 2112  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-if 2983  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-grp 9316  df-gid 9317  df-subg 9424

Copyright terms: Public domain