Theorem subgid 8120

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 8116	. . . . . . 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 795	. . . . 5 |- (H e. (SubGrp` G) -> H e. Grp)
4		eqid 1475	. . . . . 6 |- ran H = ran H
5		subgid.2	. . . . . 6 |- T = (Id` H)
6	4, 5	grpidcl 8059	. . . . 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 8118	. . . 4 |- (H e. (SubGrp` G) -> ((T e. ran H /\ T e. ran H) -> (THT) = (TGT)))
9	7, 7, 8	mp2and 703	. . 3 |- (H e. (SubGrp` G) -> (THT) = (TGT))
10	4, 5	grplid 8061	. . . 4 |- ((H e. Grp /\ T e. ran H) -> (THT) = T)
11	10, 3, 7	sylanc 471	. . 3 |- (H e. (SubGrp` G) -> (THT) = T)
12	9, 11	eqtr3d 1509	. 2 |- (H e. (SubGrp` G) -> (TGT) = T)
13		eqid 1475	. . . 4 |- ran G = ran G
14		subgid.1	. . . 4 |- U = (Id` G)
15	13, 14	grpid 8065	. . 3 |- ((G e. Grp /\ T e. ran G) -> (T = U <-> (TGT) = T))
16	2	3simp1d 794	. . 3 |- (H e. (SubGrp` G) -> G e. Grp)
17	13, 4	subgrnss 8119	. . . 4 |- (H e. (SubGrp` G) -> ran H (_ ran G)
18	17, 7	sseldd 2068	. . 3 |- (H e. (SubGrp` G) -> T e. ran G)
19	15, 16, 18	sylanc 471	. 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 775   = wceq 956   e. wcel 958   (_ wss 2047  ran crn 3171  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  Idcgi 8034  SubGrpcsubg 8114

This theorem is referenced by:  cayleylem3 10411

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-grp 8037  df-gid 8038  df-subg 8115

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