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Theorem divsinv 15001
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 14974 . . . . . 6  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
2 subgrcl 14951 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
31, 2syl 16 . . . . 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 14852 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  V )  ->  ( I `  X
)  e.  V )
73, 6sylan 459 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  (
I `  X )  e.  V )
8 divsgrp.h . . . . 5  |-  H  =  ( G  /.s  ( G ~QG  S
) )
9 eqid 2438 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
10 eqid 2438 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
118, 4, 9, 10divsadd 14999 . . . 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 1281 . . 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 2438 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
144, 9, 13, 5grprinv 14854 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  V )  ->  ( X ( +g  `  G ) ( I `
 X ) )  =  ( 0g `  G ) )
153, 14sylan 459 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  ( X ( +g  `  G
) ( I `  X ) )  =  ( 0g `  G
) )
16 eceq1 6943 . . . 4  |-  ( ( X ( +g  `  G
) ( I `  X ) )  =  ( 0g `  G
)  ->  [ ( X ( +g  `  G
) ( I `  X ) ) ] ( G ~QG  S )  =  [
( 0g `  G
) ] ( G ~QG  S ) )
1715, 16syl 16 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ ( X ( +g  `  G
) ( I `  X ) ) ] ( G ~QG  S )  =  [
( 0g `  G
) ] ( G ~QG  S ) )
188, 13divs0 15000 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  [ ( 0g `  G ) ] ( G ~QG  S )  =  ( 0g `  H ) )
1918adantr 453 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ ( 0g `  G ) ] ( G ~QG  S )  =  ( 0g `  H ) )
2012, 17, 193eqtrd 2474 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  ( [ X ] ( G ~QG  S ) ( +g  `  H
) [ ( I `
 X ) ] ( G ~QG  S ) )  =  ( 0g `  H
) )
218divsgrp 14997 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
2221adantr 453 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  H  e.  Grp )
23 eqid 2438 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
248, 4, 23divseccl 14998 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ X ] ( G ~QG  S )  e.  ( Base `  H
) )
258, 4, 23divseccl 14998 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( I `  X )  e.  V
)  ->  [ (
I `  X ) ] ( G ~QG  S )  e.  ( Base `  H
) )
267, 25syldan 458 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  X  e.  V )  ->  [ ( I `  X ) ] ( G ~QG  S )  e.  ( Base `  H
) )
27 eqid 2438 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
28 divsinv.n . . . 4  |-  N  =  ( inv g `  H )
2923, 10, 27, 28grpinvid1 14855 . . 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 1185 . 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 225 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   [cec 6905   Basecbs 13471   +g cplusg 13531   0gc0g 13725    /.s cqus 13733   Grpcgrp 14687   inv gcminusg 14688  SubGrpcsubg 14940  NrmSGrpcnsg 14941   ~QG cqg 14942
This theorem is referenced by:  divssub  15002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-ec 6909  df-qs 6913  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-0g 13729  df-imas 13736  df-divs 13737  df-mnd 14692  df-grp 14814  df-minusg 14815  df-subg 14943  df-nsg 14944  df-eqg 14945
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