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Theorem psubspi 30558
Description: Property of a projective subspace. (Contributed by NM, 13-Jan-2012.)
Hypotheses
Ref Expression
psubspset.l  |-  .<_  =  ( le `  K )
psubspset.j  |-  .\/  =  ( join `  K )
psubspset.a  |-  A  =  ( Atoms `  K )
psubspset.s  |-  S  =  ( PSubSp `  K )
Assertion
Ref Expression
psubspi  |-  ( ( ( K  e.  D  /\  X  e.  S  /\  P  e.  A
)  /\  E. q  e.  X  E. r  e.  X  P  .<_  ( q  .\/  r ) )  ->  P  e.  X )
Distinct variable groups:    A, r,
q    K, q, r    X, q, r    A, q    P, q, r
Allowed substitution hints:    D( r, q)    S( r, q)    .\/ ( r, q)    .<_ ( r, q)

Proof of Theorem psubspi
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 psubspset.l . . . . . 6  |-  .<_  =  ( le `  K )
2 psubspset.j . . . . . 6  |-  .\/  =  ( join `  K )
3 psubspset.a . . . . . 6  |-  A  =  ( Atoms `  K )
4 psubspset.s . . . . . 6  |-  S  =  ( PSubSp `  K )
51, 2, 3, 4ispsubsp2 30557 . . . . 5  |-  ( K  e.  D  ->  ( X  e.  S  <->  ( X  C_  A  /\  A. p  e.  A  ( E. q  e.  X  E. r  e.  X  p  .<_  ( q  .\/  r
)  ->  p  e.  X ) ) ) )
65simplbda 607 . . . 4  |-  ( ( K  e.  D  /\  X  e.  S )  ->  A. p  e.  A  ( E. q  e.  X  E. r  e.  X  p  .<_  ( q  .\/  r )  ->  p  e.  X ) )
76ex 423 . . 3  |-  ( K  e.  D  ->  ( X  e.  S  ->  A. p  e.  A  ( E. q  e.  X  E. r  e.  X  p  .<_  ( q  .\/  r )  ->  p  e.  X ) ) )
8 breq1 4042 . . . . . 6  |-  ( p  =  P  ->  (
p  .<_  ( q  .\/  r )  <->  P  .<_  ( q  .\/  r ) ) )
982rexbidv 2599 . . . . 5  |-  ( p  =  P  ->  ( E. q  e.  X  E. r  e.  X  p  .<_  ( q  .\/  r )  <->  E. q  e.  X  E. r  e.  X  P  .<_  ( q  .\/  r ) ) )
10 eleq1 2356 . . . . 5  |-  ( p  =  P  ->  (
p  e.  X  <->  P  e.  X ) )
119, 10imbi12d 311 . . . 4  |-  ( p  =  P  ->  (
( E. q  e.  X  E. r  e.  X  p  .<_  ( q 
.\/  r )  ->  p  e.  X )  <->  ( E. q  e.  X  E. r  e.  X  P  .<_  ( q  .\/  r )  ->  P  e.  X ) ) )
1211rspccv 2894 . . 3  |-  ( A. p  e.  A  ( E. q  e.  X  E. r  e.  X  p  .<_  ( q  .\/  r )  ->  p  e.  X )  ->  ( P  e.  A  ->  ( E. q  e.  X  E. r  e.  X  P  .<_  ( q  .\/  r )  ->  P  e.  X ) ) )
137, 12syl6 29 . 2  |-  ( K  e.  D  ->  ( X  e.  S  ->  ( P  e.  A  -> 
( E. q  e.  X  E. r  e.  X  P  .<_  ( q 
.\/  r )  ->  P  e.  X )
) ) )
14133imp1 1164 1  |-  ( ( ( K  e.  D  /\  X  e.  S  /\  P  e.  A
)  /\  E. q  e.  X  E. 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   A.wral 2556   E.wrex 2557    C_ wss 3165   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:  psubspi2N  30559  paddidm  30652
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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