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Theorem psubspi 33403
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 33402 . . . . 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 624 . . . 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 434 . . 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 4307 . . . . . 6  |-  ( p  =  P  ->  (
p  .<_  ( q  .\/  r )  <->  P  .<_  ( q  .\/  r ) ) )
982rexbidv 2770 . . . . 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 2503 . . . . 5  |-  ( p  =  P  ->  (
p  e.  X  <->  P  e.  X ) )
119, 10imbi12d 320 . . . 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 3082 . . 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 33 . 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 1200 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2727   E.wrex 2728    C_ wss 3340   class class class wbr 4304   ` cfv 5430  (class class class)co 6103   lecple 14257   joincjn 15126   Atomscatm 32920   PSubSpcpsubsp 33152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fv 5438  df-ov 6106  df-psubsp 33159
This theorem is referenced by:  psubspi2N  33404  paddidm  33497
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