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Theorem psubspi 30544
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 30543 . . . . 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 608 . . . 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 424 . . 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 4215 . . . . . 6  |-  ( p  =  P  ->  (
p  .<_  ( q  .\/  r )  <->  P  .<_  ( q  .\/  r ) ) )
982rexbidv 2748 . . . . 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 2496 . . . . 5  |-  ( p  =  P  ->  (
p  e.  X  <->  P  e.  X ) )
119, 10imbi12d 312 . . . 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 3049 . . 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 31 . 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 1166 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    C_ wss 3320   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   lecple 13536   joincjn 14401   Atomscatm 30061   PSubSpcpsubsp 30293
This theorem is referenced by:  psubspi2N  30545  paddidm  30638
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  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-rab 2714  df-v 2958  df-sbc 3162  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-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-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-psubsp 30300
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