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Theorem psubspi2N 30607
Description: Property of a projective subspace. (Contributed by NM, 13-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubspset.l  |-  .<_  =  ( le `  K )
psubspset.j  |-  .\/  =  ( join `  K )
psubspset.a  |-  A  =  ( Atoms `  K )
psubspset.s  |-  S  =  ( PSubSp `  K )
Assertion
Ref Expression
psubspi2N  |-  ( ( ( K  e.  D  /\  X  e.  S  /\  P  e.  A
)  /\  ( Q  e.  X  /\  R  e.  X  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P  e.  X )

Proof of Theorem psubspi2N
Dummy variables  r 
q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6090 . . . 4  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
21breq2d 4226 . . 3  |-  ( q  =  Q  ->  ( P  .<_  ( q  .\/  r )  <->  P  .<_  ( Q  .\/  r ) ) )
3 oveq2 6091 . . . 4  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
43breq2d 4226 . . 3  |-  ( r  =  R  ->  ( P  .<_  ( Q  .\/  r )  <->  P  .<_  ( Q  .\/  R ) ) )
52, 4rspc2ev 3062 . 2  |-  ( ( Q  e.  X  /\  R  e.  X  /\  P  .<_  ( Q  .\/  R ) )  ->  E. q  e.  X  E. r  e.  X  P  .<_  ( q  .\/  r ) )
6 psubspset.l . . 3  |-  .<_  =  ( le `  K )
7 psubspset.j . . 3  |-  .\/  =  ( join `  K )
8 psubspset.a . . 3  |-  A  =  ( Atoms `  K )
9 psubspset.s . . 3  |-  S  =  ( PSubSp `  K )
106, 7, 8, 9psubspi 30606 . 2  |-  ( ( ( K  e.  D  /\  X  e.  S  /\  P  e.  A
)  /\  E. q  e.  X  E. r  e.  X  P  .<_  ( q  .\/  r ) )  ->  P  e.  X )
115, 10sylan2 462 1  |-  ( ( ( K  e.  D  /\  X  e.  S  /\  P  e.  A
)  /\  ( Q  e.  X  /\  R  e.  X  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   lecple 13538   joincjn 14403   Atomscatm 30123   PSubSpcpsubsp 30355
This theorem is referenced by:  pclclN  30750  pclfinN  30759  pclfinclN  30809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-psubsp 30362
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