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Theorem pclbtwnN 30694
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 734 . 2  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  X  C_  ( U `  Y
) )
2 simpll 731 . . . 4  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  K  e.  V )
3 simprl 733 . . . 4  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  Y  C_  X )
4 eqid 2436 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 pclid.s . . . . . 6  |-  S  =  ( PSubSp `  K )
64, 5psubssat 30551 . . . . 5  |-  ( ( K  e.  V  /\  X  e.  S )  ->  X  C_  ( Atoms `  K ) )
76adantr 452 . . . 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 30691 . . . 4  |-  ( ( K  e.  V  /\  Y  C_  X  /\  X  C_  ( Atoms `  K )
)  ->  ( U `  Y )  C_  ( U `  X )
)
102, 3, 7, 9syl3anc 1184 . . 3  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  ( U `  Y )  C_  ( U `  X
) )
115, 8pclidN 30693 . . . 4  |-  ( ( K  e.  V  /\  X  e.  S )  ->  ( U `  X
)  =  X )
1211adantr 452 . . 3  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  ( U `  X )  =  X )
1310, 12sseqtrd 3384 . 2  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  ( U `  Y )  C_  X )
141, 13eqssd 3365 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 359    = wceq 1652    e. wcel 1725    C_ wss 3320   ` cfv 5454   Atomscatm 30061   PSubSpcpsubsp 30293   PClcpclN 30684
This theorem is referenced by:  pclfinN  30697
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
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-ral 2710  df-rex 2711  df-reu 2712  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-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  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-psubsp 30300  df-pclN 30685
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