Theorem qussub 14017

Description: Value of the group subtraction operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)

Hypotheses

Ref	Expression
qusgrp.h	|- H = ( G /.s ( G ~QG S ) )
qusinv.v	|- V = ( Base ` G )
qussub.p	|- .- = ( -g ` G )
qussub.a	|- N = ( -g ` H )

Assertion

Ref	Expression
qussub	|- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( [ X ] ( G ~QG S ) N [ Y ] ( G ~QG S ) ) = [ ( X .- Y ) ] ( G ~QG S ) )

Proof of Theorem qussub

Step	Hyp	Ref	Expression
1		qusgrp.h	. . . . 5 |- H = ( G /.s ( G ~QG S ) )
2		qusinv.v	. . . . 5 |- V = ( Base ` G )
3		eqid 2238	. . . . 5 |- ( Base ` H ) = ( Base ` H )
4	1, 2, 3	quseccl 14013	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ X ] ( G ~QG S ) e. ( Base ` H ) )
5	4	3adant3 1048	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> [ X ] ( G ~QG S ) e. ( Base ` H ) )
6	1, 2, 3	quseccl 14013	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ Y e. V ) -> [ Y ] ( G ~QG S ) e. ( Base ` H ) )
7		eqid 2238	. . . 4 |- ( +g ` H ) = ( +g ` H )
8		eqid 2238	. . . 4 |- ( invg ` H ) = ( invg ` H )
9		qussub.a	. . . 4 |- N = ( -g ` H )
10	3, 7, 8, 9	grpsubval 13828	. . 3 |- ( ( [ X ] ( G ~QG S ) e. ( Base ` H ) /\ [ Y ] ( G ~QG S ) e. ( Base ` H ) ) -> ( [ X ] ( G ~QG S ) N [ Y ] ( G ~QG S ) ) = ( [ X ] ( G ~QG S ) ( +g ` H ) ( ( invg ` H ) ` [ Y ] ( G ~QG S ) ) ) )
11	5, 6, 10	3imp3i2an 1214	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( [ X ] ( G ~QG S ) N [ Y ] ( G ~QG S ) ) = ( [ X ] ( G ~QG S ) ( +g ` H ) ( ( invg ` H ) ` [ Y ] ( G ~QG S ) ) ) )
12		eqid 2238	. . . . 5 |- ( invg ` G ) = ( invg ` G )
13	1, 2, 12, 8	qusinv 14016	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ Y e. V ) -> ( ( invg ` H ) ` [ Y ] ( G ~QG S ) ) = [ ( ( invg ` G ) ` Y ) ] ( G ~QG S ) )
14	13	3adant2 1047	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( ( invg ` H ) ` [ Y ] ( G ~QG S ) ) = [ ( ( invg ` G ) ` Y ) ] ( G ~QG S ) )
15	14	oveq2d 6091	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) ( ( invg ` H ) ` [ Y ] ( G ~QG S ) ) ) = ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( ( invg ` G ) ` Y ) ] ( G ~QG S ) ) )
16		nsgsubg 13985	. . . . . . 7 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
17		subgrcl 13959	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> G e. Grp )
18	16, 17	syl 14	. . . . . 6 |- ( S e. (NrmSGrp ` G ) -> G e. Grp )
19	2, 12	grpinvcl 13830	. . . . . 6 |- ( ( G e. Grp /\ Y e. V ) -> ( ( invg ` G ) ` Y ) e. V )
20	18, 19	sylan 283	. . . . 5 |- ( ( S e. (NrmSGrp ` G ) /\ Y e. V ) -> ( ( invg ` G ) ` Y ) e. V )
21	20	3adant2 1047	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( ( invg ` G ) ` Y ) e. V )
22		eqid 2238	. . . . 5 |- ( +g ` G ) = ( +g ` G )
23	1, 2, 22, 7	qusadd 14014	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ ( ( invg ` G ) ` Y ) e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( ( invg ` G ) ` Y ) ] ( G ~QG S ) ) = [ ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) ] ( G ~QG S ) )
24	21, 23	syld3an3 1323	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( ( invg ` G ) ` Y ) ] ( G ~QG S ) ) = [ ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) ] ( G ~QG S ) )
25		qussub.p	. . . . . 6 |- .- = ( -g ` G )
26	2, 22, 12, 25	grpsubval 13828	. . . . 5 |- ( ( X e. V /\ Y e. V ) -> ( X .- Y ) = ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) )
27	26	3adant1 1046	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( X .- Y ) = ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) )
28	27	eceq1d 6833	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> [ ( X .- Y ) ] ( G ~QG S ) = [ ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) ] ( G ~QG S ) )
29	24, 28	eqtr4d 2274	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( ( invg ` G ) ` Y ) ] ( G ~QG S ) ) = [ ( X .- Y ) ] ( G ~QG S ) )
30	11, 15, 29	3eqtrd 2275	1 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( [ X ] ( G ~QG S ) N [ Y ] ( G ~QG S ) ) = [ ( X .- Y ) ] ( G ~QG S ) )

Syntax hints:   -> wi 4   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5372  (class class class)co 6075   [cec 6795   Basecbs 13330   +g cplusg 13408   /._s cqus 13600   Grpcgrp 13782   invgcminusg 13783   -gcsg 13784  SubGrpcsubg 13947  NrmSGrpcnsg 13948   ~_QG cqg 13949

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-tp 3713  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-er 6797  df-ec 6799  df-qs 6803  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-mulr 13422  df-0g 13589  df-iimas 13601  df-qus 13602  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-sbg 13787  df-subg 13950  df-nsg 13951  df-eqg 13952

This theorem is referenced by: (None)