Theorem qussub 14040

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 14036	. . . 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 14036	. . 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 13851	. . 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 14039	. . . 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 6101	. 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 14008	. . . . . . 7 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
17		subgrcl 13982	. . . . . . 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 13853	. . . . . 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 14037	. . . 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 13851	. . . . 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 6843	. . 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   [cec 6805   Basecbs 13352   +g cplusg 13431   /._s cqus 13623   Grpcgrp 13805   invgcminusg 13806   -gcsg 13807  SubGrpcsubg 13970  NrmSGrpcnsg 13971   ~_QG cqg 13972

This proof depends on 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 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof 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 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-er 6807  df-ec 6809  df-qs 6813  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-0g 13612  df-iimas 13624  df-qus 13625  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-sbg 13810  df-subg 13973  df-nsg 13974  df-eqg 13975

This theorem is used by: (None)