Theorem subgabl 9432

Description: A subgroup of an Abelian group is Abelian. (Contributed by Paul Chapman, 25-Apr-2008.)

Assertion

Ref	Expression
subgabl	|- ((G e. Abel /\ H e. (SubGrp` G)) -> H e. Abel)

Proof of Theorem subgabl

Step	Hyp	Ref	Expression
1		simpr 350	. 2 |- ((G e. Abel /\ H e. (SubGrp` G)) -> H e. (SubGrp` G))
2		eqid 1884	. . . . . . . . . 10 |- ran G = ran G
3		eqid 1884	. . . . . . . . . 10 |- ran H = ran H
4	2, 3	subgrnss 9428	. . . . . . . . 9 |- (H e. (SubGrp` G) -> ran H C_ ran G)
5	4	sseld 2619	. . . . . . . 8 |- (H e. (SubGrp` G) -> (x e. ran H -> x e. ran G))
6	4	sseld 2619	. . . . . . . 8 |- (H e. (SubGrp` G) -> (y e. ran H -> y e. ran G))
7	5, 6	anim12d 617	. . . . . . 7 |- (H e. (SubGrp` G) -> ((x e. ran H /\ y e. ran H) -> (x e. ran G /\ y e. ran G)))
8	2	isabl 9409	. . . . . . . . 9 |- (G e. Abel <-> (G e. Grp /\ A.x e. ran GA.y e. ran G(xGy) = (yGx)))
9	8	simprbi 353	. . . . . . . 8 |- (G e. Abel -> A.x e. ran GA.y e. ran G(xGy) = (yGx))
10		ra42 2157	. . . . . . . 8 |- (A.x e. ran GA.y e. ran G(xGy) = (yGx) -> ((x e. ran G /\ y e. ran G) -> (xGy) = (yGx)))
11	9, 10	syl 12	. . . . . . 7 |- (G e. Abel -> ((x e. ran G /\ y e. ran G) -> (xGy) = (yGx)))
12	7, 11	sylan9r 519	. . . . . 6 |- ((G e. Abel /\ H e. (SubGrp` G)) -> ((x e. ran H /\ y e. ran H) -> (xGy) = (yGx)))
13	12	imp 377	. . . . 5 |- (((G e. Abel /\ H e. (SubGrp` G)) /\ (x e. ran H /\ y e. ran H)) -> (xGy) = (yGx))
14	3	subgopr 9427	. . . . . . 7 |- (H e. (SubGrp` G) -> ((x e. ran H /\ y e. ran H) -> (xHy) = (xGy)))
15	14	adantl 424	. . . . . 6 |- ((G e. Abel /\ H e. (SubGrp` G)) -> ((x e. ran H /\ y e. ran H) -> (xHy) = (xGy)))
16	15	imp 377	. . . . 5 |- (((G e. Abel /\ H e. (SubGrp` G)) /\ (x e. ran H /\ y e. ran H)) -> (xHy) = (xGy))
17	3	subgopr 9427	. . . . . . . 8 |- (H e. (SubGrp` G) -> ((y e. ran H /\ x e. ran H) -> (yHx) = (yGx)))
18	17	ancomsd 485	. . . . . . 7 |- (H e. (SubGrp` G) -> ((x e. ran H /\ y e. ran H) -> (yHx) = (yGx)))
19	18	adantl 424	. . . . . 6 |- ((G e. Abel /\ H e. (SubGrp` G)) -> ((x e. ran H /\ y e. ran H) -> (yHx) = (yGx)))
20	19	imp 377	. . . . 5 |- (((G e. Abel /\ H e. (SubGrp` G)) /\ (x e. ran H /\ y e. ran H)) -> (yHx) = (yGx))
21	13, 16, 20	3eqtr4d 1937	. . . 4 |- (((G e. Abel /\ H e. (SubGrp` G)) /\ (x e. ran H /\ y e. ran H)) -> (xHy) = (yHx))
22	21	ex 402	. . 3 |- ((G e. Abel /\ H e. (SubGrp` G)) -> ((x e. ran H /\ y e. ran H) -> (xHy) = (yHx)))
23	22	r19.21aivv 2183	. 2 |- ((G e. Abel /\ H e. (SubGrp` G)) -> A.x e. ran HA.y e. ran H(xHy) = (yHx))
24	3	isabl 9409	. . . 4 |- (H e. Abel <-> (H e. Grp /\ A.x e. ran HA.y e. ran H(xHy) = (yHx)))
25	24	biimpri 169	. . 3 |- ((H e. Grp /\ A.x e. ran HA.y e. ran H(xHy) = (yHx)) -> H e. Abel)
26		issubg 9425	. . . 4 |- (H e. (SubGrp` G) <-> (G e. Grp /\ H e. Grp /\ H C_ G))
27	26	simp2bi 892	. . 3 |- (H e. (SubGrp` G) -> H e. Grp)
28	25, 27	sylan 497	. 2 |- ((H e. (SubGrp` G) /\ A.x e. ran HA.y e. ran H(xHy) = (yHx)) -> H e. Abel)
29	1, 23, 28	syl11anc 524	1 |- ((G e. Abel /\ H e. (SubGrp` G)) -> H e. Abel)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  ran crn 3987  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Abelcabl 9407  SubGrpcsubg 9423

This theorem is referenced by:  efghgrpilem 10073

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-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-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  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-fo 4012  df-fv 4014  df-opr 4886  df-grp 9316  df-abl 9408  df-subg 9424

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