Theorem subgabl 13402

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 13248	. . 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 2193	. . . 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 12746	. 2 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> ( +g ` G ) = ( +g ` H ) )
10	1	subggrp 13247	. . 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 2193	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
15	14	subgss 13244	. . . . 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 3180	. . 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 3180	. . 3 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> y e. ( Base ` G ) )
21	14, 5	ablcom 13373	. . 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 13369	1 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> H e. Abel )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   C_ wss 3153   ` cfv 5254  (class class class)co 5918   Basecbs 12618   ↾_s cress 12619   +g cplusg 12695   Grpcgrp 13072  SubGrpcsubg 13237   Abelcabl 13355

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-ltadd 7988

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-grp 13075  df-subg 13240  df-cmn 13356  df-abl 13357

This theorem is referenced by:  issubrng2  13706  rnglidlrng  13994