Theorem subgsub 13316

Description: The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.)

Hypotheses

Ref	Expression
subgsubcl.p	|- .- = ( -g ` G )
subgsub.h	|- H = ( G ↾s S )
subgsub.n	|- N = ( -g ` H )

Assertion

Ref	Expression
subgsub	|- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X .- Y ) = ( X N Y ) )

Proof of Theorem subgsub

Step	Hyp	Ref	Expression
1		subgsub.h	. . . . . 6 |- H = ( G ↾s S )
2	1	a1i 9	. . . . 5 |- ( S e. (SubGrp ` G ) -> H = ( G ↾s S ) )
3		eqidd 2197	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
4		id 19	. . . . 5 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` G ) )
5		subgrcl 13309	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
6	2, 3, 4, 5	ressplusgd 12806	. . . 4 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` H ) )
7	6	3ad2ant1 1020	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( +g ` G ) = ( +g ` H ) )
8		eqidd 2197	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X = X )
9		eqid 2196	. . . . 5 |- ( invg ` G ) = ( invg ` G )
10		eqid 2196	. . . . 5 |- ( invg ` H ) = ( invg ` H )
11	1, 9, 10	subginv 13311	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ Y e. S ) -> ( ( invg ` G ) ` Y ) = ( ( invg ` H ) ` Y ) )
12	11	3adant2 1018	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( ( invg ` G ) ` Y ) = ( ( invg ` H ) ` Y ) )
13	7, 8, 12	oveq123d 5943	. 2 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) = ( X ( +g ` H ) ( ( invg ` H ) ` Y ) ) )
14		eqid 2196	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
15	14	subgss 13304	. . . . 5 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
16	15	3ad2ant1 1020	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> S C_ ( Base ` G ) )
17		simp2 1000	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X e. S )
18	16, 17	sseldd 3184	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X e. ( Base ` G ) )
19		simp3 1001	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> Y e. S )
20	16, 19	sseldd 3184	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> Y e. ( Base ` G ) )
21		eqid 2196	. . . 4 |- ( +g ` G ) = ( +g ` G )
22		subgsubcl.p	. . . 4 |- .- = ( -g ` G )
23	14, 21, 9, 22	grpsubval 13178	. . 3 |- ( ( X e. ( Base ` G ) /\ Y e. ( Base ` G ) ) -> ( X .- Y ) = ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) )
24	18, 20, 23	syl2anc 411	. 2 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X .- Y ) = ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) )
25	1	subgbas 13308	. . . . 5 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
26	25	3ad2ant1 1020	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> S = ( Base ` H ) )
27	17, 26	eleqtrd 2275	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X e. ( Base ` H ) )
28	19, 26	eleqtrd 2275	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> Y e. ( Base ` H ) )
29		eqid 2196	. . . 4 |- ( Base ` H ) = ( Base ` H )
30		eqid 2196	. . . 4 |- ( +g ` H ) = ( +g ` H )
31		subgsub.n	. . . 4 |- N = ( -g ` H )
32	29, 30, 10, 31	grpsubval 13178	. . 3 |- ( ( X e. ( Base ` H ) /\ Y e. ( Base ` H ) ) -> ( X N Y ) = ( X ( +g ` H ) ( ( invg ` H ) ` Y ) ) )
33	27, 28, 32	syl2anc 411	. 2 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X N Y ) = ( X ( +g ` H ) ( ( invg ` H ) ` Y ) ) )
34	13, 24, 33	3eqtr4d 2239	1 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X .- Y ) = ( X N Y ) )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2167   C_ wss 3157   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾_s cress 12679   +g cplusg 12755   Grpcgrp 13132   invgcminusg 13133   -gcsg 13134  SubGrpcsubg 13297

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-sbg 13137  df-subg 13300

This theorem is referenced by:  zringsubgval  14161  zndvds  14205