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Theorem psubspi 29912
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 29911 . . . . 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 4149 . . . . . 6  |-  ( p  =  P  ->  (
p  .<_  ( q  .\/  r )  <->  P  .<_  ( q  .\/  r ) ) )
982rexbidv 2685 . . . . 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 2440 . . . . 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 2985 . . 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 1649    e. wcel 1717   A.wral 2642   E.wrex 2643    C_ wss 3256   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   lecple 13456   joincjn 14321   Atomscatm 29429   PSubSpcpsubsp 29661
This theorem is referenced by:  psubspi2N  29913  paddidm  30006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-psubsp 29668
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