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Theorem subginv 14906
Description: The inverse of an element in a subgroup is the same as the inverse in the larger group. (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
subg0.h  |-  H  =  ( Gs  S )
subginv.i  |-  I  =  ( inv g `  G )
subginv.j  |-  J  =  ( inv g `  H )
Assertion
Ref Expression
subginv  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  (
I `  X )  =  ( J `  X ) )

Proof of Theorem subginv
StepHypRef Expression
1 subg0.h . . . . . 6  |-  H  =  ( Gs  S )
21subggrp 14902 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
32adantr 452 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  H  e.  Grp )
41subgbas 14903 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
54eleq2d 2471 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( X  e.  S  <->  X  e.  ( Base `  H ) ) )
65biimpa 471 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  X  e.  ( Base `  H
) )
7 eqid 2404 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
8 eqid 2404 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
9 eqid 2404 . . . . 5  |-  ( 0g
`  H )  =  ( 0g `  H
)
10 subginv.j . . . . 5  |-  J  =  ( inv g `  H )
117, 8, 9, 10grprinv 14807 . . . 4  |-  ( ( H  e.  Grp  /\  X  e.  ( Base `  H ) )  -> 
( X ( +g  `  H ) ( J `
 X ) )  =  ( 0g `  H ) )
123, 6, 11syl2anc 643 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( X ( +g  `  H
) ( J `  X ) )  =  ( 0g `  H
) )
13 eqid 2404 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
141, 13ressplusg 13526 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
1514adantr 452 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( +g  `  G )  =  ( +g  `  H
) )
1615oveqd 6057 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( X ( +g  `  G
) ( J `  X ) )  =  ( X ( +g  `  H ) ( J `
 X ) ) )
17 eqid 2404 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
181, 17subg0 14905 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
1918adantr 452 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( 0g `  G )  =  ( 0g `  H
) )
2012, 16, 193eqtr4d 2446 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( X ( +g  `  G
) ( J `  X ) )  =  ( 0g `  G
) )
21 subgrcl 14904 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
2221adantr 452 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  G  e.  Grp )
23 eqid 2404 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
2423subgss 14900 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
2524sselda 3308 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  X  e.  ( Base `  G
) )
267, 10grpinvcl 14805 . . . . . . . 8  |-  ( ( H  e.  Grp  /\  X  e.  ( Base `  H ) )  -> 
( J `  X
)  e.  ( Base `  H ) )
2726ex 424 . . . . . . 7  |-  ( H  e.  Grp  ->  ( X  e.  ( Base `  H )  ->  ( J `  X )  e.  ( Base `  H
) ) )
282, 27syl 16 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  ( X  e.  ( Base `  H
)  ->  ( J `  X )  e.  (
Base `  H )
) )
294eleq2d 2471 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  ( ( J `  X )  e.  S  <->  ( J `  X )  e.  (
Base `  H )
) )
3028, 5, 293imtr4d 260 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( X  e.  S  ->  ( J `
 X )  e.  S ) )
3130imp 419 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( J `  X )  e.  S )
3224sselda 3308 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( J `  X )  e.  S )  ->  ( J `  X )  e.  ( Base `  G
) )
3331, 32syldan 457 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( J `  X )  e.  ( Base `  G
) )
34 subginv.i . . . 4  |-  I  =  ( inv g `  G )
3523, 13, 17, 34grpinvid1 14808 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  ( Base `  G )  /\  ( J `  X )  e.  ( Base `  G
) )  ->  (
( I `  X
)  =  ( J `
 X )  <->  ( X
( +g  `  G ) ( J `  X
) )  =  ( 0g `  G ) ) )
3622, 25, 33, 35syl3anc 1184 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  (
( I `  X
)  =  ( J `
 X )  <->  ( X
( +g  `  G ) ( J `  X
) )  =  ( 0g `  G ) ) )
3720, 36mpbird 224 1  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  (
I `  X )  =  ( J `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   Basecbs 13424   ↾s cress 13425   +g cplusg 13484   0gc0g 13678   Grpcgrp 14640   inv gcminusg 14641  SubGrpcsubg 14893
This theorem is referenced by:  subginvcl  14908  subgsub  14911  subgmulg  14913  zlpirlem1  16723  prmirred  16730  subgtgp  18088  clmneg  19059  qrngneg  21270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-subg 14896
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