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Theorem divssub 15863
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 2454 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
41, 2, 3divseccl 15859 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ X ] ( G ~QG  S )  e.  ( Base `  H
) )
543adant3 1008 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  Y  e.  V
)  ->  [ X ] ( G ~QG  S )  e.  ( Base `  H
) )
61, 2, 3divseccl 15859 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  Y  e.  V )  ->  [ Y ] ( G ~QG  S )  e.  ( Base `  H
) )
763adant2 1007 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  Y  e.  V
)  ->  [ Y ] ( G ~QG  S )  e.  ( Base `  H
) )
8 eqid 2454 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
9 eqid 2454 . . . 4  |-  ( invg `  H )  =  ( invg `  H )
10 divssub.a . . . 4  |-  N  =  ( -g `  H
)
113, 8, 9, 10grpsubval 15703 . . 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 ) ) ) )
125, 7, 11syl2anc 661 . 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 ) ) ) )
13 eqid 2454 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
141, 2, 13, 9divsinv 15862 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  Y  e.  V )  ->  (
( invg `  H ) `  [ Y ] ( G ~QG  S ) )  =  [ ( ( invg `  G ) `  Y
) ] ( G ~QG  S ) )
15143adant2 1007 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  Y  e.  V
)  ->  ( ( invg `  H ) `
 [ Y ]
( G ~QG  S ) )  =  [ ( ( invg `  G ) `
 Y ) ] ( G ~QG  S ) )
1615oveq2d 6219 . 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 ) ) )
17 nsgsubg 15835 . . . . . . 7  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
18 subgrcl 15808 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
1917, 18syl 16 . . . . . 6  |-  ( S  e.  (NrmSGrp `  G
)  ->  G  e.  Grp )
202, 13grpinvcl 15705 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  V )  ->  ( ( invg `  G ) `  Y
)  e.  V )
2119, 20sylan 471 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  Y  e.  V )  ->  (
( invg `  G ) `  Y
)  e.  V )
22213adant2 1007 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  Y  e.  V
)  ->  ( ( invg `  G ) `
 Y )  e.  V )
23 eqid 2454 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
241, 2, 23, 8divsadd 15860 . . . 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 ) )
2522, 24syld3an3 1264 . . 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 ) )
26 divssub.p . . . . . 6  |-  .-  =  ( -g `  G )
272, 23, 13, 26grpsubval 15703 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( ( invg `  G ) `
 Y ) ) )
28273adant1 1006 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  Y  e.  V
)  ->  ( X  .-  Y )  =  ( X ( +g  `  G
) ( ( invg `  G ) `
 Y ) ) )
29 eceq1 7250 . . . 4  |-  ( ( X  .-  Y )  =  ( X ( +g  `  G ) ( ( invg `  G ) `  Y
) )  ->  [ ( X  .-  Y ) ] ( G ~QG  S )  =  [ ( X ( +g  `  G
) ( ( invg `  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 ) ( ( invg `  G
) `  Y )
) ] ( G ~QG  S ) )
3125, 30eqtr4d 2498 . 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 ) )
3212, 16, 313eqtrd 2499 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 setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5529  (class class class)co 6203   [cec 7212   Basecbs 14295   +g cplusg 14360    /.s cqus 14565   Grpcgrp 15532   invgcminusg 15533   -gcsg 15535  SubGrpcsubg 15797  NrmSGrpcnsg 15798   ~QG cqg 15799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-ec 7216  df-qs 7220  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-fz 11558  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-mulr 14374  df-sca 14376  df-vsca 14377  df-ip 14378  df-tset 14379  df-ple 14380  df-ds 14382  df-0g 14502  df-imas 14568  df-divs 14569  df-mnd 15537  df-grp 15667  df-minusg 15668  df-sbg 15669  df-subg 15800  df-nsg 15801  df-eqg 15802
This theorem is referenced by:  divstgplem  19826
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