Theorem subgcl 13635

Description: A subgroup is closed under group operation. (Contributed by Mario Carneiro, 2-Dec-2014.)

Hypothesis

Ref	Expression
subgcl.p	|- .+ = ( +g ` G )

Assertion

Ref	Expression
subgcl	|- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X .+ Y ) e. S )

Proof of Theorem subgcl

Step	Hyp	Ref	Expression
1		eqid 2207	. . 3 |- ( Base ` ( G ↾s S ) ) = ( Base ` ( G ↾s S ) )
2		eqid 2207	. . 3 |- ( +g ` ( G ↾s S ) ) = ( +g ` ( G ↾s S ) )
3		eqid 2207	. . . . 5 |- ( G ↾s S ) = ( G ↾s S )
4	3	subggrp 13628	. . . 4 |- ( S e. (SubGrp ` G ) -> ( G ↾s S ) e. Grp )
5	4	3ad2ant1 1021	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( G ↾s S ) e. Grp )
6		simp2 1001	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X e. S )
7	3	subgbas 13629	. . . . 5 |- ( S e. (SubGrp ` G ) -> S = ( Base ` ( G ↾s S ) ) )
8	7	3ad2ant1 1021	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> S = ( Base ` ( G ↾s S ) ) )
9	6, 8	eleqtrd 2286	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X e. ( Base ` ( G ↾s S ) ) )
10		simp3 1002	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> Y e. S )
11	10, 8	eleqtrd 2286	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> Y e. ( Base ` ( G ↾s S ) ) )
12	1, 2, 5, 9, 11	grpcld 13461	. 2 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X ( +g ` ( G ↾s S ) ) Y ) e. ( Base ` ( G ↾s S ) ) )
13		eqidd 2208	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( G ↾s S ) = ( G ↾s S ) )
14		subgcl.p	. . . . . 6 |- .+ = ( +g ` G )
15	14	a1i 9	. . . . 5 |- ( S e. (SubGrp ` G ) -> .+ = ( +g ` G ) )
16		id 19	. . . . 5 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` G ) )
17		subgrcl 13630	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
18	13, 15, 16, 17	ressplusgd 13076	. . . 4 |- ( S e. (SubGrp ` G ) -> .+ = ( +g ` ( G ↾s S ) ) )
19	18	3ad2ant1 1021	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> .+ = ( +g ` ( G ↾s S ) ) )
20	19	oveqd 5984	. 2 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X .+ Y ) = ( X ( +g ` ( G ↾s S ) ) Y ) )
21	12, 20, 8	3eltr4d 2291	1 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X .+ Y ) e. S )

Syntax hints:   -> wi 4   /\ w3a 981   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   Basecbs 12947   ↾_s cress 12948   +g cplusg 13024   Grpcgrp 13447  SubGrpcsubg 13618

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-subg 13621

This theorem is referenced by:  subgsubcl  13636  subgmulgcl  13638  issubg2m  13640  subgintm  13649  ssnmz  13662  eqger  13675  eqgcpbl  13679  resghm  13711  ghmpreima  13717  subrngacl  14085  subrgacl  14109  islss4  14259  dflidl2rng  14358