Theorem subgabl 14226

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 14040	. . 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 2238	. . . 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 13542	. 2 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> ( +g ` G ) = ( +g ` H ) )
10	1	subggrp 14039	. . 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 1052	. . 3 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> G e. Abel )
13		simp1r 1053	. . . . 5 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> S e. (SubGrp ` G ) )
14		eqid 2238	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
15	14	subgss 14036	. . . . 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 1029	. . . 4 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> x e. S )
18	16, 17	sseldd 3249	. . 3 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> x e. ( Base ` G ) )
19		simp3 1030	. . . 4 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> y e. S )
20	16, 19	sseldd 3249	. . 3 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> y e. ( Base ` G ) )
21	14, 5	ablcom 14196	. . 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 1278	. 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 14192	1 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> H e. Abel )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   ` cfv 5380  (class class class)co 6088   Basecbs 13410   ↾_s cress 13411   +g cplusg 13490   Grpcgrp 13864  SubGrpcsubg 14029   Abelcabl 14178

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-iress 13418  df-plusg 13503  df-grp 13867  df-subg 14032  df-cmn 14179  df-abl 14180

This theorem is used by:  issubrng2  14627  rnglidlrng  14944