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Theorem psubspi2N 30559
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 5881 . . . 4  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
21breq2d 4051 . . 3  |-  ( q  =  Q  ->  ( P  .<_  ( q  .\/  r )  <->  P  .<_  ( Q  .\/  r ) ) )
3 oveq2 5882 . . . 4  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
43breq2d 4051 . . 3  |-  ( r  =  R  ->  ( P  .<_  ( Q  .\/  r )  <->  P  .<_  ( Q  .\/  R ) ) )
52, 4rspc2ev 2905 . 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 30558 . 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 460 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   Atomscatm 30075   PSubSpcpsubsp 30307
This theorem is referenced by:  pclclN  30702  pclfinN  30711  pclfinclN  30761
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  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-rab 2565  df-v 2803  df-sbc 3005  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-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-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-psubsp 30314
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