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Theorem divsinv 14676
Description: Value of the group inverse 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
)
divsinv.i  |-  I  =  ( inv g `  G )
divsinv.n  |-  N  =  ( inv g `  H )
Assertion
Ref Expression
divsinv  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  ( N `  [ X ] ( G ~QG  S ) )  =  [ ( I `  X ) ] ( G ~QG  S ) )

Proof of Theorem divsinv
StepHypRef Expression
1 nsgsubg 14649 . . . . . 6  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
2 subgrcl 14626 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
31, 2syl 15 . . . . 5  |-  ( S  e.  (NrmSGrp `  G
)  ->  G  e.  Grp )
4 divsinv.v . . . . . 6  |-  V  =  ( Base `  G
)
5 divsinv.i . . . . . 6  |-  I  =  ( inv g `  G )
64, 5grpinvcl 14527 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  V )  ->  ( I `  X
)  e.  V )
73, 6sylan 457 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  (
I `  X )  e.  V )
8 divsgrp.h . . . . 5  |-  H  =  ( G  /.s  ( G ~QG  S
) )
9 eqid 2283 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
10 eqid 2283 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
118, 4, 9, 10divsadd 14674 . . . 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 1278 . . 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 2283 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
144, 9, 13, 5grprinv 14529 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  V )  ->  ( X ( +g  `  G ) ( I `
 X ) )  =  ( 0g `  G ) )
153, 14sylan 457 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  ( X ( +g  `  G
) ( I `  X ) )  =  ( 0g `  G
) )
16 eceq1 6696 . . . 4  |-  ( ( X ( +g  `  G
) ( I `  X ) )  =  ( 0g `  G
)  ->  [ ( X ( +g  `  G
) ( I `  X ) ) ] ( G ~QG  S )  =  [
( 0g `  G
) ] ( G ~QG  S ) )
1715, 16syl 15 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ ( X ( +g  `  G
) ( I `  X ) ) ] ( G ~QG  S )  =  [
( 0g `  G
) ] ( G ~QG  S ) )
188, 13divs0 14675 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  [ ( 0g `  G ) ] ( G ~QG  S )  =  ( 0g `  H ) )
1918adantr 451 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ ( 0g `  G ) ] ( G ~QG  S )  =  ( 0g `  H ) )
2012, 17, 193eqtrd 2319 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  ( [ X ] ( G ~QG  S ) ( +g  `  H
) [ ( I `
 X ) ] ( G ~QG  S ) )  =  ( 0g `  H
) )
218divsgrp 14672 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
2221adantr 451 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  H  e.  Grp )
23 eqid 2283 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
248, 4, 23divseccl 14673 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ X ] ( G ~QG  S )  e.  ( Base `  H
) )
258, 4, 23divseccl 14673 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( I `  X )  e.  V
)  ->  [ (
I `  X ) ] ( G ~QG  S )  e.  ( Base `  H
) )
267, 25syldan 456 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ ( I `  X ) ] ( G ~QG  S )  e.  ( Base `  H
) )
27 eqid 2283 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
28 divsinv.n . . . 4  |-  N  =  ( inv g `  H )
2923, 10, 27, 28grpinvid1 14530 . . 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
) ) )
3022, 24, 26, 29syl3anc 1182 . 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 ) ) )
3120, 30mpbird 223 1  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  ( N `  [ X ] ( G ~QG  S ) )  =  [ ( I `  X ) ] ( G ~QG  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   [cec 6658   Basecbs 13148   +g cplusg 13208   0gc0g 13400    /.s cqus 13408   Grpcgrp 14362   inv gcminusg 14363  SubGrpcsubg 14615  NrmSGrpcnsg 14616   ~QG cqg 14617
This theorem is referenced by:  divssub  14677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-imas 13411  df-divs 13412  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-nsg 14619  df-eqg 14620
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