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Theorem subginv 16077
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 16073 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
32adantr 465 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  H  e.  Grp )
41subgbas 16074 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
54eleq2d 2511 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( X  e.  S  <->  X  e.  ( Base `  H ) ) )
65biimpa 484 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  X  e.  ( Base `  H
) )
7 eqid 2441 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
8 eqid 2441 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
9 eqid 2441 . . . . 5  |-  ( 0g
`  H )  =  ( 0g `  H
)
10 subginv.j . . . . 5  |-  J  =  ( invg `  H )
117, 8, 9, 10grprinv 15966 . . . 4  |-  ( ( H  e.  Grp  /\  X  e.  ( Base `  H ) )  -> 
( X ( +g  `  H ) ( J `
 X ) )  =  ( 0g `  H ) )
123, 6, 11syl2anc 661 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( X ( +g  `  H
) ( J `  X ) )  =  ( 0g `  H
) )
13 eqid 2441 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
141, 13ressplusg 14611 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
1514adantr 465 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( +g  `  G )  =  ( +g  `  H
) )
1615oveqd 6294 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( X ( +g  `  G
) ( J `  X ) )  =  ( X ( +g  `  H ) ( J `
 X ) ) )
17 eqid 2441 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
181, 17subg0 16076 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
1918adantr 465 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( 0g `  G )  =  ( 0g `  H
) )
2012, 16, 193eqtr4d 2492 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( X ( +g  `  G
) ( J `  X ) )  =  ( 0g `  G
) )
21 subgrcl 16075 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
2221adantr 465 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  G  e.  Grp )
23 eqid 2441 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
2423subgss 16071 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
2524sselda 3486 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  X  e.  ( Base `  G
) )
267, 10grpinvcl 15964 . . . . . . . 8  |-  ( ( H  e.  Grp  /\  X  e.  ( Base `  H ) )  -> 
( J `  X
)  e.  ( Base `  H ) )
2726ex 434 . . . . . . 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 2511 . . . . . 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 429 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( J `  X )  e.  S )
3224sselda 3486 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  ( J `  X )  e.  S )  ->  ( J `  X )  e.  ( Base `  G
) )
3331, 32syldan 470 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( J `  X )  e.  ( Base `  G
) )
34 subginv.i . . . 4  |-  I  =  ( invg `  G )
3523, 13, 17, 34grpinvid1 15967 . . 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 1227 . 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 369    = wceq 1381    e. wcel 1802   ` cfv 5574  (class class class)co 6277   Basecbs 14504   ↾s cress 14505   +g cplusg 14569   0gc0g 14709   Grpcgrp 15922   invgcminusg 15923  SubGrpcsubg 16064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-minusg 15927  df-subg 16067
This theorem is referenced by:  subginvcl  16079  subgsub  16082  subgmulg  16084  zringlpirlem1  18376  zlpirlem1  18381  prmirred  18392  prmirredOLD  18395  psgninv  18485  subgtgp  20470  clmneg  21447  qrngneg  23673
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