Theorem subgabl 8119

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		eqid 1478	. . . . 5 |- ran H = ran H
2	1	isabl 8097	. . . 4 |- (H e. Abel <-> (H e. Grp /\ A.x e. ran HA.y e. ran H(xHy) = (yHx)))
3	2	biimpr 152	. . 3 |- ((H e. Grp /\ A.x e. ran HA.y e. ran H(xHy) = (yHx)) -> H e. Abel)
4		issubg 8112	. . . . 5 |- (H e. (SubGrp` G) <-> (G e. Grp /\ H e. Grp /\ H (_ G))
5	4	biimp 151	. . . 4 |- (H e. (SubGrp` G) -> (G e. Grp /\ H e. Grp /\ H (_ G))
6	5	3simp2d 797	. . 3 |- (H e. (SubGrp` G) -> H e. Grp)
7	3, 6	sylan 450	. 2 |- ((H e. (SubGrp` G) /\ A.x e. ran HA.y e. ran H(xHy) = (yHx)) -> H e. Abel)
8		pm3.27 323	. 2 |- ((G e. Abel /\ H e. (SubGrp` G)) -> H e. (SubGrp` G))
9		eqid 1478	. . . . . . . . . 10 |- ran G = ran G
10	9, 1	subgrnss 8115	. . . . . . . . 9 |- (H e. (SubGrp` G) -> ran H (_ ran G)
11	10	sseld 2070	. . . . . . . 8 |- (H e. (SubGrp` G) -> (x e. ran H -> x e. ran G))
12	10	sseld 2070	. . . . . . . 8 |- (H e. (SubGrp` G) -> (y e. ran H -> y e. ran G))
13	11, 12	anim12d 560	. . . . . . 7 |- (H e. (SubGrp` G) -> ((x e. ran H /\ y e. ran H) -> (x e. ran G /\ y e. ran G)))
14	9	isabl 8097	. . . . . . . . 9 |- (G e. Abel <-> (G e. Grp /\ A.x e. ran GA.y e. ran G(xGy) = (yGx)))
15	14	pm3.27bi 326	. . . . . . . 8 |- (G e. Abel -> A.x e. ran GA.y e. ran G(xGy) = (yGx))
16		ra42 1699	. . . . . . . 8 |- (A.x e. ran GA.y e. ran G(xGy) = (yGx) -> ((x e. ran G /\ y e. ran G) -> (xGy) = (yGx)))
17	15, 16	syl 10	. . . . . . 7 |- (G e. Abel -> ((x e. ran G /\ y e. ran G) -> (xGy) = (yGx)))
18	13, 17	sylan9r 471	. . . . . 6 |- ((G e. Abel /\ H e. (SubGrp` G)) -> ((x e. ran H /\ y e. ran H) -> (xGy) = (yGx)))
19	18	imp 350	. . . . 5 |- (((G e. Abel /\ H e. (SubGrp` G)) /\ (x e. ran H /\ y e. ran H)) -> (xGy) = (yGx))
20	1	subgopr 8114	. . . . . . 7 |- (H e. (SubGrp` G) -> ((x e. ran H /\ y e. ran H) -> (xHy) = (xGy)))
21	20	adantl 390	. . . . . 6 |- ((G e. Abel /\ H e. (SubGrp` G)) -> ((x e. ran H /\ y e. ran H) -> (xHy) = (xGy)))
22	21	imp 350	. . . . 5 |- (((G e. Abel /\ H e. (SubGrp` G)) /\ (x e. ran H /\ y e. ran H)) -> (xHy) = (xGy))
23	1	subgopr 8114	. . . . . . . 8 |- (H e. (SubGrp` G) -> ((y e. ran H /\ x e. ran H) -> (yHx) = (yGx)))
24	23	ancomsd 439	. . . . . . 7 |- (H e. (SubGrp` G) -> ((x e. ran H /\ y e. ran H) -> (yHx) = (yGx)))
25	24	adantl 390	. . . . . 6 |- ((G e. Abel /\ H e. (SubGrp` G)) -> ((x e. ran H /\ y e. ran H) -> (yHx) = (yGx)))
26	25	imp 350	. . . . 5 |- (((G e. Abel /\ H e. (SubGrp` G)) /\ (x e. ran H /\ y e. ran H)) -> (yHx) = (yGx))
27	19, 22, 26	3eqtr4d 1520	. . . 4 |- (((G e. Abel /\ H e. (SubGrp` G)) /\ (x e. ran H /\ y e. ran H)) -> (xHy) = (yHx))
28	27	ex 373	. . 3 |- ((G e. Abel /\ H e. (SubGrp` G)) -> ((x e. ran H /\ y e. ran H) -> (xHy) = (yHx)))
29	28	r19.21aivv 1723	. 2 |- ((G e. Abel /\ H e. (SubGrp` G)) -> A.x e. ran HA.y e. ran H(xHy) = (yHx))
30	7, 8, 29	sylanc 473	1 |- ((G e. Abel /\ H e. (SubGrp` G)) -> H e. Abel)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648   (_ wss 2050  ran crn 3177  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  Abelcabl 8095  SubGrpcsubg 8110

This theorem is referenced by:  efghgrpilem 8714

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-rab 1655  df-v 1815  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-abl 8096  df-subg 8111

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