Theorem subgabl 13294

Description: A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.)

Hypothesis

Ref	Expression
subgabl.h	|- H = ( G ↾s S )

Assertion

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

Proof of Theorem subgabl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subgabl.h	. . . 4 |- H = ( G ↾s S )
2	1	subgbas 13142	. . 3 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
3	2	adantl 277	. 2 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> S = ( Base ` H ) )
4	1	a1i 9	. . 3 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> H = ( G ↾s S ) )
5		eqid 2189	. . . 4 |- ( +g ` G ) = ( +g ` G )
6	5	a1i 9	. . 3 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> ( +g ` G ) = ( +g ` G ) )
7		simpr 110	. . 3 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> S e. (SubGrp ` G ) )
8		simpl 109	. . 3 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> G e. Abel )
9	4, 6, 7, 8	ressplusgd 12651	. 2 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> ( +g ` G ) = ( +g ` H ) )
10	1	subggrp 13141	. . 3 |- ( S e. (SubGrp ` G ) -> H e. Grp )
11	10	adantl 277	. 2 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> H e. Grp )
12		simp1l 1023	. . 3 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> G e. Abel )
13		simp1r 1024	. . . . 5 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> S e. (SubGrp ` G ) )
14		eqid 2189	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
15	14	subgss 13138	. . . . 5 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
16	13, 15	syl 14	. . . 4 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> S C_ ( Base ` G ) )
17		simp2 1000	. . . 4 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> x e. S )
18	16, 17	sseldd 3171	. . 3 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> x e. ( Base ` G ) )
19		simp3 1001	. . . 4 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> y e. S )
20	16, 19	sseldd 3171	. . 3 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> y e. ( Base ` G ) )
21	14, 5	ablcom 13267	. . 3 |- ( ( G e. Abel /\ x e. ( Base ` G ) /\ y e. ( Base ` G ) ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
22	12, 18, 20, 21	syl3anc 1249	. 2 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
23	3, 9, 11, 22	isabld 13263	1 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> H e. Abel )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160   C_ wss 3144   ` cfv 5238  (class class class)co 5900   Basecbs 12523   ↾_s cress 12524   +g cplusg 12600   Grpcgrp 12968  SubGrpcsubg 13131   Abelcabl 13249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-addcom 7946  ax-addass 7948  ax-i2m1 7951  ax-0lt1 7952  ax-0id 7954  ax-rnegex 7955  ax-pre-ltirr 7958  ax-pre-ltadd 7962

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-pnf 8029  df-mnf 8030  df-ltxr 8032  df-inn 8955  df-2 9013  df-ndx 12526  df-slot 12527  df-base 12529  df-sets 12530  df-iress 12531  df-plusg 12613  df-grp 12971  df-subg 13134  df-cmn 13250  df-abl 13251

This theorem is referenced by:  issubrng2  13582  rnglidlrng  13839