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Theorem pclbtwnN 30086
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 2283 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 pclid.s . . . . . 6  |-  S  =  ( PSubSp `  K )
64, 5psubssat 29943 . . . . 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 30083 . . . 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 30085 . . . 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 3214 . 2  |-  ( ( ( K  e.  V  /\  X  e.  S
)  /\  ( Y  C_  X  /\  X  C_  ( U `  Y ) ) )  ->  ( U `  Y )  C_  X )
141, 13eqssd 3196 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 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255   Atomscatm 29453   PSubSpcpsubsp 29685   PClcpclN 30076
This theorem is referenced by:  pclfinN  30089
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-rep 4131  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-int 3863  df-iun 3907  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-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-psubsp 29692  df-pclN 30077
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