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Theorem subgsub 16085
Description: The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.)
Hypotheses
Ref Expression
subgsubcl.p  |-  .-  =  ( -g `  G )
subgsub.h  |-  H  =  ( Gs  S )
subgsub.n  |-  N  =  ( -g `  H
)
Assertion
Ref Expression
subgsub  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X  .-  Y )  =  ( X N Y ) )

Proof of Theorem subgsub
StepHypRef Expression
1 subgsub.h . . . . 5  |-  H  =  ( Gs  S )
2 eqid 2467 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2ressplusg 14614 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
433ad2ant1 1017 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  ( +g  `  G )  =  ( +g  `  H
) )
5 eqidd 2468 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  X  =  X )
6 eqid 2467 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
7 eqid 2467 . . . . 5  |-  ( invg `  H )  =  ( invg `  H )
81, 6, 7subginv 16080 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  Y  e.  S )  ->  (
( invg `  G ) `  Y
)  =  ( ( invg `  H
) `  Y )
)
983adant2 1015 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  (
( invg `  G ) `  Y
)  =  ( ( invg `  H
) `  Y )
)
104, 5, 9oveq123d 6316 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X ( +g  `  G
) ( ( invg `  G ) `
 Y ) )  =  ( X ( +g  `  H ) ( ( invg `  H ) `  Y
) ) )
11 eqid 2467 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1211subgss 16074 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
13123ad2ant1 1017 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  S  C_  ( Base `  G
) )
14 simp2 997 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  X  e.  S )
1513, 14sseldd 3510 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  X  e.  ( Base `  G
) )
16 simp3 998 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  Y  e.  S )
1713, 16sseldd 3510 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  Y  e.  ( Base `  G
) )
18 subgsubcl.p . . . 4  |-  .-  =  ( -g `  G )
1911, 2, 6, 18grpsubval 15965 . . 3  |-  ( ( X  e.  ( Base `  G )  /\  Y  e.  ( Base `  G
) )  ->  ( X  .-  Y )  =  ( X ( +g  `  G ) ( ( invg `  G
) `  Y )
) )
2015, 17, 19syl2anc 661 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X  .-  Y )  =  ( X ( +g  `  G ) ( ( invg `  G
) `  Y )
) )
211subgbas 16077 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
22213ad2ant1 1017 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  S  =  ( Base `  H
) )
2314, 22eleqtrd 2557 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  X  e.  ( Base `  H
) )
2416, 22eleqtrd 2557 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  Y  e.  ( Base `  H
) )
25 eqid 2467 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
26 eqid 2467 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
27 subgsub.n . . . 4  |-  N  =  ( -g `  H
)
2825, 26, 7, 27grpsubval 15965 . . 3  |-  ( ( X  e.  ( Base `  H )  /\  Y  e.  ( Base `  H
) )  ->  ( X N Y )  =  ( X ( +g  `  H ) ( ( invg `  H
) `  Y )
) )
2923, 24, 28syl2anc 661 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X N Y )  =  ( X ( +g  `  H ) ( ( invg `  H
) `  Y )
) )
3010, 20, 293eqtr4d 2518 1  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X  .-  Y )  =  ( X N Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3481   ` cfv 5594  (class class class)co 6295   Basecbs 14507   ↾s cress 14508   +g cplusg 14572   invgcminusg 15926   -gcsg 15927  SubGrpcsubg 16067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070
This theorem is referenced by:  zndvds  18457  resubgval  18514  frlmsubgval  18667  scmatsgrp1  18893  subgngp  21017  clmsub  21448  qqhucn  27798  zringsubgval  32477
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