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

Theorem subgsub 16412
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 2454 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2ressplusg 14830 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
433ad2ant1 1015 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  ( +g  `  G )  =  ( +g  `  H
) )
5 eqidd 2455 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  X  =  X )
6 eqid 2454 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
7 eqid 2454 . . . . 5  |-  ( invg `  H )  =  ( invg `  H )
81, 6, 7subginv 16407 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  Y  e.  S )  ->  (
( invg `  G ) `  Y
)  =  ( ( invg `  H
) `  Y )
)
983adant2 1013 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  (
( invg `  G ) `  Y
)  =  ( ( invg `  H
) `  Y )
)
104, 5, 9oveq123d 6291 . 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 2454 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1211subgss 16401 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
13123ad2ant1 1015 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  S  C_  ( Base `  G
) )
14 simp2 995 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  X  e.  S )
1513, 14sseldd 3490 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  X  e.  ( Base `  G
) )
16 simp3 996 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  Y  e.  S )
1713, 16sseldd 3490 . . 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 16292 . . 3  |-  ( ( X  e.  ( Base `  G )  /\  Y  e.  ( Base `  G
) )  ->  ( X  .-  Y )  =  ( X ( +g  `  G ) ( ( invg `  G
) `  Y )
) )
2015, 17, 19syl2anc 659 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X  .-  Y )  =  ( X ( +g  `  G ) ( ( invg `  G
) `  Y )
) )
211subgbas 16404 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
22213ad2ant1 1015 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  S  =  ( Base `  H
) )
2314, 22eleqtrd 2544 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  X  e.  ( Base `  H
) )
2416, 22eleqtrd 2544 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  Y  e.  ( Base `  H
) )
25 eqid 2454 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
26 eqid 2454 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
27 subgsub.n . . . 4  |-  N  =  ( -g `  H
)
2825, 26, 7, 27grpsubval 16292 . . 3  |-  ( ( X  e.  ( Base `  H )  /\  Y  e.  ( Base `  H
) )  ->  ( X N Y )  =  ( X ( +g  `  H ) ( ( invg `  H
) `  Y )
) )
2923, 24, 28syl2anc 659 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X N Y )  =  ( X ( +g  `  H ) ( ( invg `  H
) `  Y )
) )
3010, 20, 293eqtr4d 2505 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 971    = wceq 1398    e. wcel 1823    C_ wss 3461   ` cfv 5570  (class class class)co 6270   Basecbs 14716   ↾s cress 14717   +g cplusg 14784   invgcminusg 16253   -gcsg 16254  SubGrpcsubg 16394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397
This theorem is referenced by:  zndvds  18761  resubgval  18818  frlmsubgval  18969  scmatsgrp1  19191  subgngp  21315  clmsub  21746  qqhucn  28207  zringsubgval  33249
  Copyright terms: Public domain W3C validator