Theorem qussub 13904

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 2231	. . . . 5 |- ( Base ` H ) = ( Base ` H )
4	1, 2, 3	quseccl 13900	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ X ] ( G ~QG S ) e. ( Base ` H ) )
5	4	3adant3 1044	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> [ X ] ( G ~QG S ) e. ( Base ` H ) )
6	1, 2, 3	quseccl 13900	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ Y e. V ) -> [ Y ] ( G ~QG S ) e. ( Base ` H ) )
7		eqid 2231	. . . 4 |- ( +g ` H ) = ( +g ` H )
8		eqid 2231	. . . 4 |- ( invg ` H ) = ( invg ` H )
9		qussub.a	. . . 4 |- N = ( -g ` H )
10	3, 7, 8, 9	grpsubval 13709	. . 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 1210	. 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 2231	. . . . 5 |- ( invg ` G ) = ( invg ` G )
13	1, 2, 12, 8	qusinv 13903	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ Y e. V ) -> ( ( invg ` H ) ` [ Y ] ( G ~QG S ) ) = [ ( ( invg ` G ) ` Y ) ] ( G ~QG S ) )
14	13	3adant2 1043	. . 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 6044	. 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 13872	. . . . . . 7 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
17		subgrcl 13846	. . . . . . 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 13711	. . . . . 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 1043	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( ( invg ` G ) ` Y ) e. V )
22		eqid 2231	. . . . 5 |- ( +g ` G ) = ( +g ` G )
23	1, 2, 22, 7	qusadd 13901	. . . 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 1319	. . 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 13709	. . . . 5 |- ( ( X e. V /\ Y e. V ) -> ( X .- Y ) = ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) )
27	26	3adant1 1042	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( X .- Y ) = ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) )
28	27	eceq1d 6781	. . 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 2267	. 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 2268	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 1005   = wceq 1398   e. wcel 2202   ` cfv 5333  (class class class)co 6028   [cec 6743   Basecbs 13162   +g cplusg 13240   /._s cqus 13463   Grpcgrp 13663   invgcminusg 13664   -gcsg 13665  SubGrpcsubg 13834  NrmSGrpcnsg 13835   ~_QG cqg 13836

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-er 6745  df-ec 6747  df-qs 6751  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-mulr 13254  df-0g 13421  df-iimas 13465  df-qus 13466  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-sbg 13668  df-subg 13837  df-nsg 13838  df-eqg 13839

This theorem is referenced by: (None)