Theorem subgsub 13877

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 2233	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
4		id 19	. . . . 5 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` G ) )
5		subgrcl 13870	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
6	2, 3, 4, 5	ressplusgd 13316	. . . 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 2233	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X = X )
9		eqid 2232	. . . . 5 |- ( invg ` G ) = ( invg ` G )
10		eqid 2232	. . . . 5 |- ( invg ` H ) = ( invg ` H )
11	1, 9, 10	subginv 13872	. . . 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 6062	. 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 2232	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
15	14	subgss 13865	. . . . 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 3238	. . 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 3238	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> Y e. ( Base ` G ) )
21		eqid 2232	. . . 4 |- ( +g ` G ) = ( +g ` G )
22		subgsubcl.p	. . . 4 |- .- = ( -g ` G )
23	14, 21, 9, 22	grpsubval 13733	. . 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 13869	. . . . 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 2311	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> X e. ( Base ` H ) )
28	19, 26	eleqtrd 2311	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> Y e. ( Base ` H ) )
29		eqid 2232	. . . 4 |- ( Base ` H ) = ( Base ` H )
30		eqid 2232	. . . 4 |- ( +g ` H ) = ( +g ` H )
31		subgsub.n	. . . 4 |- N = ( -g ` H )
32	29, 30, 10, 31	grpsubval 13733	. . 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 2275	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 2203   C_ wss 3210   ` cfv 5343  (class class class)co 6041   Basecbs 13186   ↾_s cress 13187   +g cplusg 13264   Grpcgrp 13687   invgcminusg 13688   -gcsg 13689  SubGrpcsubg 13858

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4218  ax-sep 4221  ax-pow 4279  ax-pr 4314  ax-un 4545  ax-setind 4650  ax-cnex 8206  ax-resscn 8207  ax-1cn 8208  ax-1re 8209  ax-icn 8210  ax-addcl 8211  ax-addrcl 8212  ax-mulcl 8213  ax-addcom 8215  ax-addass 8217  ax-i2m1 8220  ax-0lt1 8221  ax-0id 8223  ax-rnegex 8224  ax-pre-ltirr 8227  ax-pre-ltadd 8231

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3506  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-int 3943  df-iun 3986  df-br 4103  df-opab 4165  df-mpt 4166  df-id 4405  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-res 4752  df-ima 4753  df-iota 5303  df-fun 5345  df-fn 5346  df-f 5347  df-f1 5348  df-fo 5349  df-f1o 5350  df-fv 5351  df-riota 5994  df-ov 6044  df-oprab 6045  df-mpo 6046  df-1st 6325  df-2nd 6326  df-pnf 8298  df-mnf 8299  df-ltxr 8301  df-inn 9226  df-2 9284  df-ndx 13189  df-slot 13190  df-base 13192  df-sets 13193  df-iress 13194  df-plusg 13277  df-0g 13445  df-mgm 13543  df-sgrp 13589  df-mnd 13604  df-grp 13690  df-minusg 13691  df-sbg 13692  df-subg 13861

This theorem is referenced by:  zringsubgval  14725  zndvds  14769