Theorem subgabl 8123

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 1475	. . . . 5 |- ran H = ran H
2	1	isabl 8101	. . . 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 8116	. . . . 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 795	. . 3 |- (H e. (SubGrp` G) -> H e. Grp)
7	3, 6	sylan 448	. 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 1475	. . . . . . . . . 10 |- ran G = ran G
10	9, 1	subgrnss 8119	. . . . . . . . 9 |- (H e. (SubGrp` G) -> ran H (_ ran G)
11	10	sseld 2067	. . . . . . . 8 |- (H e. (SubGrp` G) -> (x e. ran H -> x e. ran G))
12	10	sseld 2067	. . . . . . . 8 |- (H e. (SubGrp` G) -> (y e. ran H -> y e. ran G))
13	11, 12	anim12d 558	. . . . . . 7 |- (H e. (SubGrp` G) -> ((x e. ran H /\ y e. ran H) -> (x e. ran G /\ y e. ran G)))
14	9	isabl 8101	. . . . . . . . 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 1696	. . . . . . . 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 469	. . . . . 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 8118	. . . . . . 7 |- (H e. (SubGrp` G) -> ((x e. ran H /\ y e. ran H) -> (xHy) = (xGy)))
21	20	adantl 388	. . . . . 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 8118	. . . . . . . 8 |- (H e. (SubGrp` G) -> ((y e. ran H /\ x e. ran H) -> (yHx) = (yGx)))
24	23	ancomsd 437	. . . . . . 7 |- (H e. (SubGrp` G) -> ((x e. ran H /\ y e. ran H) -> (yHx) = (yGx)))
25	24	adantl 388	. . . . . 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 1517	. . . 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 1720	. 2 |- ((G e. Abel /\ H e. (SubGrp` G)) -> A.x e. ran HA.y e. ran H(xHy) = (yHx))
30	7, 8, 29	sylanc 471	1 |- ((G e. Abel /\ H e. (SubGrp` G)) -> H e. Abel)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047  ran crn 3171  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  Abelcabl 8099  SubGrpcsubg 8114

This theorem is referenced by:  efghgrpilem 8719

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-rab 1652  df-v 1812  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-abl 8100  df-subg 8115

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