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Theorem divsinv 14692
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 14665 . . . . . 6  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
2 subgrcl 14642 . . . . . 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 14543 . . . . 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 2296 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
10 eqid 2296 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
118, 4, 9, 10divsadd 14690 . . . 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 2296 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
144, 9, 13, 5grprinv 14545 . . . . 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 6712 . . . 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 14691 . . . 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 2332 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  ( [ X ] ( G ~QG  S ) ( +g  `  H
) [ ( I `
 X ) ] ( G ~QG  S ) )  =  ( 0g `  H
) )
218divsgrp 14688 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
2221adantr 451 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  H  e.  Grp )
23 eqid 2296 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
248, 4, 23divseccl 14689 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ X ] ( G ~QG  S )  e.  ( Base `  H
) )
258, 4, 23divseccl 14689 . . . 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 2296 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
28 divsinv.n . . . 4  |-  N  =  ( inv g `  H )
2923, 10, 27, 28grpinvid1 14546 . . 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 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   [cec 6674   Basecbs 13164   +g cplusg 13224   0gc0g 13416    /.s cqus 13424   Grpcgrp 14378   inv gcminusg 14379  SubGrpcsubg 14631  NrmSGrpcnsg 14632   ~QG cqg 14633
This theorem is referenced by:  divssub  14693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-imas 13427  df-divs 13428  df-mnd 14383  df-grp 14505  df-minusg 14506  df-subg 14634  df-nsg 14635  df-eqg 14636
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