Theorem subgabl 14085

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 13931	. . 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 2234	. . . 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 13426	. 2 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> ( +g ` G ) = ( +g ` H ) )
10	1	subggrp 13930	. . 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 1048	. . 3 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> G e. Abel )
13		simp1r 1049	. . . . 5 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> S e. (SubGrp ` G ) )
14		eqid 2234	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
15	14	subgss 13927	. . . . 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 1025	. . . 4 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> x e. S )
18	16, 17	sseldd 3243	. . 3 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> x e. ( Base ` G ) )
19		simp3 1026	. . . 4 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> y e. S )
20	16, 19	sseldd 3243	. . 3 |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ x e. S /\ y e. S ) -> y e. ( Base ` G ) )
21	14, 5	ablcom 14056	. . 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 1274	. 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 14052	1 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> H e. Abel )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   C_ wss 3214   ` cfv 5357  (class class class)co 6058   Basecbs 13296   ↾_s cress 13297   +g cplusg 13374   Grpcgrp 13755  SubGrpcsubg 13920   Abelcabl 14038

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-grp 13758  df-subg 13923  df-cmn 14039  df-abl 14040

This theorem is referenced by:  issubrng2  14456  rnglidlrng  14772