Theorem subgsub 13774

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 2232	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
4		id 19	. . . . 5 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` G ) )
5		subgrcl 13767	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
6	2, 3, 4, 5	ressplusgd 13213	. . . 4 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` H ) )
7	6	3ad2ant1 1044	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( +g ` G ) = ( +g ` H ) )
8		eqidd 2232	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X = X )
9		eqid 2231	. . . . 5 |- ( invg ` G ) = ( invg ` G )
10		eqid 2231	. . . . 5 |- ( invg ` H ) = ( invg ` H )
11	1, 9, 10	subginv 13769	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ Y e. S ) -> ( ( invg ` G ) ` Y ) = ( ( invg ` H ) ` Y ) )
12	11	3adant2 1042	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( ( invg ` G ) ` Y ) = ( ( invg ` H ) ` Y ) )
13	7, 8, 12	oveq123d 6039	. 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 2231	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
15	14	subgss 13762	. . . . 5 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
16	15	3ad2ant1 1044	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> S C_ ( Base ` G ) )
17		simp2 1024	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X e. S )
18	16, 17	sseldd 3228	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X e. ( Base ` G ) )
19		simp3 1025	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> Y e. S )
20	16, 19	sseldd 3228	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> Y e. ( Base ` G ) )
21		eqid 2231	. . . 4 |- ( +g ` G ) = ( +g ` G )
22		subgsubcl.p	. . . 4 |- .- = ( -g ` G )
23	14, 21, 9, 22	grpsubval 13630	. . 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 13766	. . . . 5 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
26	25	3ad2ant1 1044	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> S = ( Base ` H ) )
27	17, 26	eleqtrd 2310	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X e. ( Base ` H ) )
28	19, 26	eleqtrd 2310	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> Y e. ( Base ` H ) )
29		eqid 2231	. . . 4 |- ( Base ` H ) = ( Base ` H )
30		eqid 2231	. . . 4 |- ( +g ` H ) = ( +g ` H )
31		subgsub.n	. . . 4 |- N = ( -g ` H )
32	29, 30, 10, 31	grpsubval 13630	. . 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 2274	1 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X .- Y ) = ( X N Y ) )

Syntax hints:   -> wi 4   /\ w3a 1004   = wceq 1397   e. wcel 2202   C_ wss 3200   ` cfv 5326  (class class class)co 6018   Basecbs 13083   ↾_s cress 13084   +g cplusg 13161   Grpcgrp 13584   invgcminusg 13585   -gcsg 13586  SubGrpcsubg 13755

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-ndx 13086  df-slot 13087  df-base 13089  df-sets 13090  df-iress 13091  df-plusg 13174  df-0g 13342  df-mgm 13440  df-sgrp 13486  df-mnd 13501  df-grp 13587  df-minusg 13588  df-sbg 13589  df-subg 13758

This theorem is referenced by:  zringsubgval  14621  zndvds  14665