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Theorem divssub 14929
Description: Value of the group subtraction operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
Hypotheses
Ref Expression
divsgrp.h  |-  H  =  ( G  /.s  ( G ~QG  S
) )
divsinv.v  |-  V  =  ( Base `  G
)
divssub.p  |-  .-  =  ( -g `  G )
divssub.a  |-  N  =  ( -g `  H
)
Assertion
Ref Expression
divssub  |-  ( ( 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 divssub
StepHypRef Expression
1 divsgrp.h . . . . 5  |-  H  =  ( G  /.s  ( G ~QG  S
) )
2 divsinv.v . . . . 5  |-  V  =  ( Base `  G
)
3 eqid 2389 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
41, 2, 3divseccl 14925 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ X ] ( G ~QG  S )  e.  ( Base `  H
) )
543adant3 977 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  Y  e.  V
)  ->  [ X ] ( G ~QG  S )  e.  ( Base `  H
) )
61, 2, 3divseccl 14925 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  Y  e.  V )  ->  [ Y ] ( G ~QG  S )  e.  ( Base `  H
) )
763adant2 976 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  Y  e.  V
)  ->  [ Y ] ( G ~QG  S )  e.  ( Base `  H
) )
8 eqid 2389 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
9 eqid 2389 . . . 4  |-  ( inv g `  H )  =  ( inv g `  H )
10 divssub.a . . . 4  |-  N  =  ( -g `  H
)
113, 8, 9, 10grpsubval 14777 . . 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 ) ( ( inv g `  H
) `  [ Y ] ( G ~QG  S ) ) ) )
125, 7, 11syl2anc 643 . 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 ) ( ( inv g `  H
) `  [ Y ] ( G ~QG  S ) ) ) )
13 eqid 2389 . . . . 5  |-  ( inv g `  G )  =  ( inv g `  G )
141, 2, 13, 9divsinv 14928 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  Y  e.  V )  ->  (
( inv g `  H ) `  [ Y ] ( G ~QG  S ) )  =  [ ( ( inv g `  G ) `  Y
) ] ( G ~QG  S ) )
15143adant2 976 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  Y  e.  V
)  ->  ( ( inv g `  H ) `
 [ Y ]
( G ~QG  S ) )  =  [ ( ( inv g `  G ) `
 Y ) ] ( G ~QG  S ) )
1615oveq2d 6038 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  Y  e.  V
)  ->  ( [ X ] ( G ~QG  S ) ( +g  `  H
) ( ( inv g `  H ) `
 [ Y ]
( G ~QG  S ) ) )  =  ( [ X ] ( G ~QG  S ) ( +g  `  H
) [ ( ( inv g `  G
) `  Y ) ] ( G ~QG  S ) ) )
17 nsgsubg 14901 . . . . . . 7  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
18 subgrcl 14878 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
1917, 18syl 16 . . . . . 6  |-  ( S  e.  (NrmSGrp `  G
)  ->  G  e.  Grp )
202, 13grpinvcl 14779 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  V )  ->  ( ( inv g `  G ) `  Y
)  e.  V )
2119, 20sylan 458 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  Y  e.  V )  ->  (
( inv g `  G ) `  Y
)  e.  V )
22213adant2 976 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  Y  e.  V
)  ->  ( ( inv g `  G ) `
 Y )  e.  V )
23 eqid 2389 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
241, 2, 23, 8divsadd 14926 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  ( ( inv g `  G ) `
 Y )  e.  V )  ->  ( [ X ] ( G ~QG  S ) ( +g  `  H
) [ ( ( inv g `  G
) `  Y ) ] ( G ~QG  S ) )  =  [ ( X ( +g  `  G
) ( ( inv g `  G ) `
 Y ) ) ] ( G ~QG  S ) )
2522, 24syld3an3 1229 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  Y  e.  V
)  ->  ( [ X ] ( G ~QG  S ) ( +g  `  H
) [ ( ( inv g `  G
) `  Y ) ] ( G ~QG  S ) )  =  [ ( X ( +g  `  G
) ( ( inv g `  G ) `
 Y ) ) ] ( G ~QG  S ) )
26 divssub.p . . . . . 6  |-  .-  =  ( -g `  G )
272, 23, 13, 26grpsubval 14777 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( ( inv g `  G ) `
 Y ) ) )
28273adant1 975 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  Y  e.  V
)  ->  ( X  .-  Y )  =  ( X ( +g  `  G
) ( ( inv g `  G ) `
 Y ) ) )
29 eceq1 6879 . . . 4  |-  ( ( X  .-  Y )  =  ( X ( +g  `  G ) ( ( inv g `  G ) `  Y
) )  ->  [ ( X  .-  Y ) ] ( G ~QG  S )  =  [ ( X ( +g  `  G
) ( ( inv g `  G ) `
 Y ) ) ] ( G ~QG  S ) )
3028, 29syl 16 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  Y  e.  V
)  ->  [ ( X  .-  Y ) ] ( G ~QG  S )  =  [
( X ( +g  `  G ) ( ( inv g `  G
) `  Y )
) ] ( G ~QG  S ) )
3125, 30eqtr4d 2424 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  Y  e.  V
)  ->  ( [ X ] ( G ~QG  S ) ( +g  `  H
) [ ( ( inv g `  G
) `  Y ) ] ( G ~QG  S ) )  =  [ ( X  .-  Y ) ] ( G ~QG  S ) )
3212, 16, 313eqtrd 2425 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 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5396  (class class class)co 6022   [cec 6841   Basecbs 13398   +g cplusg 13458    /.s cqus 13660   Grpcgrp 14614   inv gcminusg 14615   -gcsg 14617  SubGrpcsubg 14867  NrmSGrpcnsg 14868   ~QG cqg 14869
This theorem is referenced by:  divstgplem  18073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-ec 6845  df-qs 6849  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-fz 10978  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-sca 13474  df-vsca 13475  df-tset 13477  df-ple 13478  df-ds 13480  df-0g 13656  df-imas 13663  df-divs 13664  df-mnd 14619  df-grp 14741  df-minusg 14742  df-sbg 14743  df-subg 14870  df-nsg 14871  df-eqg 14872
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