Theorem qussub 13990

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 2234	. . . . 5 |- ( Base ` H ) = ( Base ` H )
4	1, 2, 3	quseccl 13986	. . . 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 13986	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ Y e. V ) -> [ Y ] ( G ~QG S ) e. ( Base ` H ) )
7		eqid 2234	. . . 4 |- ( +g ` H ) = ( +g ` H )
8		eqid 2234	. . . 4 |- ( invg ` H ) = ( invg ` H )
9		qussub.a	. . . 4 |- N = ( -g ` H )
10	3, 7, 8, 9	grpsubval 13801	. . 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 2234	. . . . 5 |- ( invg ` G ) = ( invg ` G )
13	1, 2, 12, 8	qusinv 13989	. . . 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 6074	. 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 13958	. . . . . . 7 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
17		subgrcl 13932	. . . . . . 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 13803	. . . . . 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 2234	. . . . 5 |- ( +g ` G ) = ( +g ` G )
23	1, 2, 22, 7	qusadd 13987	. . . 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 13801	. . . . 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 6816	. . 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 2270	. 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 2271	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 2205   ` cfv 5357  (class class class)co 6058   [cec 6778   Basecbs 13296   +g cplusg 13374   /._s cqus 13566   Grpcgrp 13755   invgcminusg 13756   -gcsg 13757  SubGrpcsubg 13920  NrmSGrpcnsg 13921   ~_QG cqg 13922

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-er 6780  df-ec 6782  df-qs 6786  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-iimas 13567  df-qus 13568  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-sbg 13760  df-subg 13923  df-nsg 13924  df-eqg 13925

This theorem is referenced by: (None)