Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pclbtwnN Unicode version

Theorem pclbtwnN 30708
Description: A projective subspace sandwiched between a set of atoms and the set's projective subspace closure equals the closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclid.s  |-  S  =  ( PSubSp `  K )
pclid.c  |-  U  =  ( PCl `  K
)
Assertion
Ref Expression
pclbtwnN  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  X  =  ( U `  Y ) )

Proof of Theorem pclbtwnN
StepHypRef Expression
1 simprr 733 . 2  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  X  C_  ( U `  Y
) )
2 simpll 730 . . . 4  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  K  e.  V )
3 simprl 732 . . . 4  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  Y  C_  X )
4 eqid 2296 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 pclid.s . . . . . 6  |-  S  =  ( PSubSp `  K )
64, 5psubssat 30565 . . . . 5  |-  ( ( K  e.  V  /\  X  e.  S )  ->  X  C_  ( Atoms `  K ) )
76adantr 451 . . . 4  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  X  C_  ( Atoms `  K )
)
8 pclid.c . . . . 5  |-  U  =  ( PCl `  K
)
94, 8pclssN 30705 . . . 4  |-  ( ( K  e.  V  /\  Y  C_  X  /\  X  C_  ( Atoms `  K )
)  ->  ( U `  Y )  C_  ( U `  X )
)
102, 3, 7, 9syl3anc 1182 . . 3  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  ( U `  Y )  C_  ( U `  X
) )
115, 8pclidN 30707 . . . 4  |-  ( ( K  e.  V  /\  X  e.  S )  ->  ( U `  X
)  =  X )
1211adantr 451 . . 3  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  ( U `  X )  =  X )
1310, 12sseqtrd 3227 . 2  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  ( U `  Y )  C_  X )
141, 13eqssd 3209 1  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  X  =  ( U `  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271   Atomscatm 30075   PSubSpcpsubsp 30307   PClcpclN 30698
This theorem is referenced by:  pclfinN  30711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-psubsp 30314  df-pclN 30699
  Copyright terms: Public domain W3C validator