MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subginv Structured version   Unicode version

Theorem subginv 16325
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  =  ( invg `  G )
subginv.j  |-  J  =  ( invg `  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 16321 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
32adantr 463 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  H  e.  Grp )
41subgbas 16322 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
54eleq2d 2452 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( X  e.  S  <->  X  e.  ( Base `  H ) ) )
65biimpa 482 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  X  e.  ( Base `  H
) )
7 eqid 2382 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
8 eqid 2382 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
9 eqid 2382 . . . . 5  |-  ( 0g
`  H )  =  ( 0g `  H
)
10 subginv.j . . . . 5  |-  J  =  ( invg `  H )
117, 8, 9, 10grprinv 16214 . . . 4  |-  ( ( H  e.  Grp  /\  X  e.  ( Base `  H ) )  -> 
( X ( +g  `  H ) ( J `
 X ) )  =  ( 0g `  H ) )
123, 6, 11syl2anc 659 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( X ( +g  `  H
) ( J `  X ) )  =  ( 0g `  H
) )
13 eqid 2382 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
141, 13ressplusg 14748 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
1514adantr 463 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( +g  `  G )  =  ( +g  `  H
) )
1615oveqd 6213 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( X ( +g  `  G
) ( J `  X ) )  =  ( X ( +g  `  H ) ( J `
 X ) ) )
17 eqid 2382 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
181, 17subg0 16324 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
1918adantr 463 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( 0g `  G )  =  ( 0g `  H
) )
2012, 16, 193eqtr4d 2433 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( X ( +g  `  G
) ( J `  X ) )  =  ( 0g `  G
) )
21 subgrcl 16323 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
2221adantr 463 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  G  e.  Grp )
23 eqid 2382 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
2423subgss 16319 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
2524sselda 3417 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  X  e.  ( Base `  G
) )
267, 10grpinvcl 16212 . . . . . . . 8  |-  ( ( H  e.  Grp  /\  X  e.  ( Base `  H ) )  -> 
( J `  X
)  e.  ( Base `  H ) )
2726ex 432 . . . . . . 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 2452 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  ( ( J `  X )  e.  S  <->  ( J `  X )  e.  (
Base `  H )
) )
3028, 5, 293imtr4d 268 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( X  e.  S  ->  ( J `
 X )  e.  S ) )
3130imp 427 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( J `  X )  e.  S )
3224sselda 3417 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( J `  X )  e.  S )  ->  ( J `  X )  e.  ( Base `  G
) )
3331, 32syldan 468 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( J `  X )  e.  ( Base `  G
) )
34 subginv.i . . . 4  |-  I  =  ( invg `  G )
3523, 13, 17, 34grpinvid1 16215 . . 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 1226 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  (
( I `  X
)  =  ( J `
 X )  <->  ( X
( +g  `  G ) ( J `  X
) )  =  ( 0g `  G ) ) )
3720, 36mpbird 232 1  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  (
I `  X )  =  ( J `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   ` cfv 5496  (class class class)co 6196   Basecbs 14634   ↾s cress 14635   +g cplusg 14702   0gc0g 14847   Grpcgrp 16170   invgcminusg 16171  SubGrpcsubg 16312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-subg 16315
This theorem is referenced by:  subginvcl  16327  subgsub  16330  subgmulg  16332  zringlpirlem1  18615  prmirred  18625  psgninv  18709  subgtgp  20689  clmneg  21666  qrngneg  23925
  Copyright terms: Public domain W3C validator