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Theorem qusinv 16134
Description: Value of the group inverse 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
)
qusinv.i  |-  I  =  ( invg `  G )
qusinv.n  |-  N  =  ( invg `  H )
Assertion
Ref Expression
qusinv  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  ( N `  [ X ] ( G ~QG  S ) )  =  [ ( I `  X ) ] ( G ~QG  S ) )

Proof of Theorem qusinv
StepHypRef Expression
1 nsgsubg 16107 . . . . . 6  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
2 subgrcl 16080 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
31, 2syl 16 . . . . 5  |-  ( S  e.  (NrmSGrp `  G
)  ->  G  e.  Grp )
4 qusinv.v . . . . . 6  |-  V  =  ( Base `  G
)
5 qusinv.i . . . . . 6  |-  I  =  ( invg `  G )
64, 5grpinvcl 15969 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  V )  ->  ( I `  X
)  e.  V )
73, 6sylan 471 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  (
I `  X )  e.  V )
8 qusgrp.h . . . . 5  |-  H  =  ( G  /.s  ( G ~QG  S
) )
9 eqid 2443 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
10 eqid 2443 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
118, 4, 9, 10qusadd 16132 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V  /\  ( I `  X )  e.  V
)  ->  ( [ X ] ( G ~QG  S ) ( +g  `  H
) [ ( I `
 X ) ] ( G ~QG  S ) )  =  [ ( X ( +g  `  G ) ( I `  X
) ) ] ( G ~QG  S ) )
127, 11mpd3an3 1326 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  ( [ X ] ( G ~QG  S ) ( +g  `  H
) [ ( I `
 X ) ] ( G ~QG  S ) )  =  [ ( X ( +g  `  G ) ( I `  X
) ) ] ( G ~QG  S ) )
13 eqid 2443 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
144, 9, 13, 5grprinv 15971 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  V )  ->  ( X ( +g  `  G ) ( I `
 X ) )  =  ( 0g `  G ) )
153, 14sylan 471 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  ( X ( +g  `  G
) ( I `  X ) )  =  ( 0g `  G
) )
1615eceq1d 7350 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ ( X ( +g  `  G
) ( I `  X ) ) ] ( G ~QG  S )  =  [
( 0g `  G
) ] ( G ~QG  S ) )
178, 13qus0 16133 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  [ ( 0g `  G ) ] ( G ~QG  S )  =  ( 0g `  H ) )
1817adantr 465 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ ( 0g `  G ) ] ( G ~QG  S )  =  ( 0g `  H ) )
1912, 16, 183eqtrd 2488 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  ( [ X ] ( G ~QG  S ) ( +g  `  H
) [ ( I `
 X ) ] ( G ~QG  S ) )  =  ( 0g `  H
) )
208qusgrp 16130 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
2120adantr 465 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  H  e.  Grp )
22 eqid 2443 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
238, 4, 22quseccl 16131 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ X ] ( G ~QG  S )  e.  ( Base `  H
) )
248, 4, 22quseccl 16131 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( I `  X )  e.  V
)  ->  [ (
I `  X ) ] ( G ~QG  S )  e.  ( Base `  H
) )
257, 24syldan 470 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ ( I `  X ) ] ( G ~QG  S )  e.  ( Base `  H
) )
26 eqid 2443 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
27 qusinv.n . . . 4  |-  N  =  ( invg `  H )
2822, 10, 26, 27grpinvid1 15972 . . 3  |-  ( ( H  e.  Grp  /\  [ X ] ( G ~QG  S )  e.  ( Base `  H )  /\  [
( I `  X
) ] ( G ~QG  S )  e.  ( Base `  H ) )  -> 
( ( N `  [ X ] ( G ~QG  S ) )  =  [
( I `  X
) ] ( G ~QG  S )  <->  ( [ X ] ( G ~QG  S ) ( +g  `  H
) [ ( I `
 X ) ] ( G ~QG  S ) )  =  ( 0g `  H
) ) )
2921, 23, 25, 28syl3anc 1229 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  (
( N `  [ X ] ( G ~QG  S ) )  =  [ ( I `  X ) ] ( G ~QG  S )  <-> 
( [ X ]
( G ~QG  S ) ( +g  `  H ) [ ( I `  X ) ] ( G ~QG  S ) )  =  ( 0g
`  H ) ) )
3019, 29mpbird 232 1  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  ( N `  [ X ] ( G ~QG  S ) )  =  [ ( I `  X ) ] ( G ~QG  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   [cec 7311   Basecbs 14509   +g cplusg 14574   0gc0g 14714    /.s cqus 14779   Grpcgrp 15927   invgcminusg 15928  SubGrpcsubg 16069  NrmSGrpcnsg 16070   ~QG cqg 16071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-ec 7315  df-qs 7319  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-fz 11682  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-0g 14716  df-imas 14782  df-qus 14783  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-subg 16072  df-nsg 16073  df-eqg 16074
This theorem is referenced by:  qussub  16135
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