Theorem subgid 8116

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 8112	. . . . . . 7 |- (H e. (SubGrp` G) <-> (G e. Grp /\ H e. Grp /\ H (_ G))
2	1	biimp 151	. . . . . 6 |- (H e. (SubGrp` G) -> (G e. Grp /\ H e. Grp /\ H (_ G))
3	2	3simp2d 797	. . . . 5 |- (H e. (SubGrp` G) -> H e. Grp)
4		eqid 1478	. . . . . 6 |- ran H = ran H
5		subgid.2	. . . . . 6 |- T = (Id` H)
6	4, 5	grpidcl 8055	. . . . 5 |- (H e. Grp -> T e. ran H)
7	3, 6	syl 10	. . . 4 |- (H e. (SubGrp` G) -> T e. ran H)
8	4	subgopr 8114	. . . 4 |- (H e. (SubGrp` G) -> ((T e. ran H /\ T e. ran H) -> (THT) = (TGT)))
9	7, 7, 8	mp2and 705	. . 3 |- (H e. (SubGrp` G) -> (THT) = (TGT))
10	4, 5	grplid 8057	. . . 4 |- ((H e. Grp /\ T e. ran H) -> (THT) = T)
11	10, 3, 7	sylanc 473	. . 3 |- (H e. (SubGrp` G) -> (THT) = T)
12	9, 11	eqtr3d 1512	. 2 |- (H e. (SubGrp` G) -> (TGT) = T)
13		eqid 1478	. . . 4 |- ran G = ran G
14		subgid.1	. . . 4 |- U = (Id` G)
15	13, 14	grpid 8061	. . 3 |- ((G e. Grp /\ T e. ran G) -> (T = U <-> (TGT) = T))
16	2	3simp1d 796	. . 3 |- (H e. (SubGrp` G) -> G e. Grp)
17	13, 4	subgrnss 8115	. . . 4 |- (H e. (SubGrp` G) -> ran H (_ ran G)
18	17, 7	sseldd 2071	. . 3 |- (H e. (SubGrp` G) -> T e. ran G)
19	15, 16, 18	sylanc 473	. 2 |- (H e. (SubGrp` G) -> (T = U <-> (TGT) = T))
20	12, 19	mpbird 196	1 |- (H e. (SubGrp` G) -> T = U)

Syntax hints:   -> wi 3   <-> wb 146   /\ w3a 777   = wceq 958   e. wcel 960   (_ wss 2050  ran crn 3177  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  Idcgi 8031  SubGrpcsubg 8110

This theorem is referenced by:  cayleylem3 10406

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-grp 8034  df-gid 8035  df-subg 8111

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