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

Theorem issubg 14621
Description: The subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypothesis
Ref Expression
issubg.b  |-  B  =  ( Base `  G
)
Assertion
Ref Expression
issubg  |-  ( S  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) )

Proof of Theorem issubg
Dummy variables  w  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-subg 14618 . . . 4  |- SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e. 
Grp } )
21dmmptss 5169 . . 3  |-  dom SubGrp  C_  Grp
3 elfvdm 5554 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  dom SubGrp )
42, 3sseldi 3178 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
5 simp1 955 . 2  |-  ( ( G  e.  Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp )  ->  G  e.  Grp )
6 fveq2 5525 . . . . . . . . . 10  |-  ( w  =  G  ->  ( Base `  w )  =  ( Base `  G
) )
7 issubg.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
86, 7syl6eqr 2333 . . . . . . . . 9  |-  ( w  =  G  ->  ( Base `  w )  =  B )
98pweqd 3630 . . . . . . . 8  |-  ( w  =  G  ->  ~P ( Base `  w )  =  ~P B )
10 oveq1 5865 . . . . . . . . 9  |-  ( w  =  G  ->  (
ws  s )  =  ( Gs  s ) )
1110eleq1d 2349 . . . . . . . 8  |-  ( w  =  G  ->  (
( ws  s )  e. 
Grp 
<->  ( Gs  s )  e. 
Grp ) )
129, 11rabeqbidv 2783 . . . . . . 7  |-  ( w  =  G  ->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }  =  { s  e. 
~P B  |  ( Gs  s )  e.  Grp } )
13 fvex 5539 . . . . . . . . . 10  |-  ( Base `  G )  e.  _V
147, 13eqeltri 2353 . . . . . . . . 9  |-  B  e. 
_V
1514pwex 4193 . . . . . . . 8  |-  ~P B  e.  _V
1615rabex 4165 . . . . . . 7  |-  { s  e.  ~P B  | 
( Gs  s )  e. 
Grp }  e.  _V
1712, 1, 16fvmpt 5602 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  =  {
s  e.  ~P B  |  ( Gs  s )  e.  Grp } )
1817eleq2d 2350 . . . . 5  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  S  e.  { s  e.  ~P B  | 
( Gs  s )  e. 
Grp } ) )
19 oveq2 5866 . . . . . . . 8  |-  ( s  =  S  ->  ( Gs  s )  =  ( Gs  S ) )
2019eleq1d 2349 . . . . . . 7  |-  ( s  =  S  ->  (
( Gs  s )  e. 
Grp 
<->  ( Gs  S )  e.  Grp ) )
2120elrab 2923 . . . . . 6  |-  ( S  e.  { s  e. 
~P B  |  ( Gs  s )  e.  Grp }  <-> 
( S  e.  ~P B  /\  ( Gs  S )  e.  Grp ) )
2214elpw2 4175 . . . . . . 7  |-  ( S  e.  ~P B  <->  S  C_  B
)
2322anbi1i 676 . . . . . 6  |-  ( ( S  e.  ~P B  /\  ( Gs  S )  e.  Grp ) 
<->  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) )
2421, 23bitri 240 . . . . 5  |-  ( S  e.  { s  e. 
~P B  |  ( Gs  s )  e.  Grp }  <-> 
( S  C_  B  /\  ( Gs  S )  e.  Grp ) )
2518, 24syl6bb 252 . . . 4  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) ) )
26 ibar 490 . . . 4  |-  ( G  e.  Grp  ->  (
( S  C_  B  /\  ( Gs  S )  e.  Grp ) 
<->  ( G  e.  Grp  /\  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) ) ) )
2725, 26bitrd 244 . . 3  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  ( S  C_  B  /\  ( Gs  S )  e.  Grp )
) ) )
28 3anass 938 . . 3  |-  ( ( G  e.  Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) 
<->  ( G  e.  Grp  /\  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) ) )
2927, 28syl6bbr 254 . 2  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) ) )
304, 5, 29pm5.21nii 342 1  |-  ( S  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   dom cdm 4689   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   Grpcgrp 14362  SubGrpcsubg 14615
This theorem is referenced by:  subgss  14622  subgid  14623  subggrp  14624  subgrcl  14626  issubg2  14636  subsubg  14640  opprsubg  15418  subrgsubg  15551  cphsubrglem  18613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-subg 14618
  Copyright terms: Public domain W3C validator