Theorem subgsub 13920

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 2235	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
4		id 19	. . . . 5 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` G ) )
5		subgrcl 13913	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
6	2, 3, 4, 5	ressplusgd 13359	. . . 4 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` H ) )
7	6	3ad2ant1 1045	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( +g ` G ) = ( +g ` H ) )
8		eqidd 2235	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X = X )
9		eqid 2234	. . . . 5 |- ( invg ` G ) = ( invg ` G )
10		eqid 2234	. . . . 5 |- ( invg ` H ) = ( invg ` H )
11	1, 9, 10	subginv 13915	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ Y e. S ) -> ( ( invg ` G ) ` Y ) = ( ( invg ` H ) ` Y ) )
12	11	3adant2 1043	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( ( invg ` G ) ` Y ) = ( ( invg ` H ) ` Y ) )
13	7, 8, 12	oveq123d 6073	. 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 2234	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
15	14	subgss 13908	. . . . 5 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
16	15	3ad2ant1 1045	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> S C_ ( Base ` G ) )
17		simp2 1025	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X e. S )
18	16, 17	sseldd 3241	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X e. ( Base ` G ) )
19		simp3 1026	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> Y e. S )
20	16, 19	sseldd 3241	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> Y e. ( Base ` G ) )
21		eqid 2234	. . . 4 |- ( +g ` G ) = ( +g ` G )
22		subgsubcl.p	. . . 4 |- .- = ( -g ` G )
23	14, 21, 9, 22	grpsubval 13776	. . 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 13912	. . . . 5 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
26	25	3ad2ant1 1045	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> S = ( Base ` H ) )
27	17, 26	eleqtrd 2313	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X e. ( Base ` H ) )
28	19, 26	eleqtrd 2313	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> Y e. ( Base ` H ) )
29		eqid 2234	. . . 4 |- ( Base ` H ) = ( Base ` H )
30		eqid 2234	. . . 4 |- ( +g ` H ) = ( +g ` H )
31		subgsub.n	. . . 4 |- N = ( -g ` H )
32	29, 30, 10, 31	grpsubval 13776	. . 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 2277	1 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X .- Y ) = ( X N Y ) )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2205   C_ wss 3213   ` cfv 5354  (class class class)co 6052   Basecbs 13229   ↾_s cress 13230   +g cplusg 13307   Grpcgrp 13730   invgcminusg 13731   -gcsg 13732  SubGrpcsubg 13901

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-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-ltadd 8245

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-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-iress 13237  df-plusg 13320  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-grp 13733  df-minusg 13734  df-sbg 13735  df-subg 13904

This theorem is referenced by:  zringsubgval  14770  zndvds  14814