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Theorem subginv 14951
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 14947 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  H  e.  Grp )
32adantr 452 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  H  e.  Grp )
41subgbas 14948 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
54eleq2d 2503 . . . . 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 2436 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
8 eqid 2436 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
9 eqid 2436 . . . . 5  |-  ( 0g
`  H )  =  ( 0g `  H
)
10 subginv.j . . . . 5  |-  J  =  ( inv g `  H )
117, 8, 9, 10grprinv 14852 . . . 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 2436 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
141, 13ressplusg 13571 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
1514adantr 452 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( +g  `  G )  =  ( +g  `  H
) )
1615oveqd 6098 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( X ( +g  `  G
) ( J `  X ) )  =  ( X ( +g  `  H ) ( J `
 X ) ) )
17 eqid 2436 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
181, 17subg0 14950 . . . 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 2478 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( X ( +g  `  G
) ( J `  X ) )  =  ( 0g `  G
) )
21 subgrcl 14949 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
2221adantr 452 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  G  e.  Grp )
23 eqid 2436 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
2423subgss 14945 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
2524sselda 3348 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  X  e.  ( Base `  G
) )
267, 10grpinvcl 14850 . . . . . . . 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 2503 . . . . . 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 3348 . . . 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 14853 . . 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 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685   inv gcminusg 14686  SubGrpcsubg 14938
This theorem is referenced by:  subginvcl  14953  subgsub  14956  subgmulg  14958  zlpirlem1  16768  prmirred  16775  subgtgp  18135  clmneg  19106  qrngneg  21317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-subg 14941
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