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Theorem ispsubsp 34942
Description: The predicate "is a projective subspace". (Contributed by NM, 2-Oct-2011.)
Hypotheses
Ref Expression
psubspset.l  |-  .<_  =  ( le `  K )
psubspset.j  |-  .\/  =  ( join `  K )
psubspset.a  |-  A  =  ( Atoms `  K )
psubspset.s  |-  S  =  ( PSubSp `  K )
Assertion
Ref Expression
ispsubsp  |-  ( K  e.  D  ->  ( X  e.  S  <->  ( X  C_  A  /\  A. p  e.  X  A. q  e.  X  A. r  e.  A  ( r  .<_  ( p  .\/  q
)  ->  r  e.  X ) ) ) )
Distinct variable groups:    A, r    q, p, r, K    X, p, q, r
Allowed substitution hints:    A( q, p)    D( r, q, p)    S( r, q, p)    .\/ ( r, q, p)    .<_ ( r, q, p)

Proof of Theorem ispsubsp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 psubspset.l . . . 4  |-  .<_  =  ( le `  K )
2 psubspset.j . . . 4  |-  .\/  =  ( join `  K )
3 psubspset.a . . . 4  |-  A  =  ( Atoms `  K )
4 psubspset.s . . . 4  |-  S  =  ( PSubSp `  K )
51, 2, 3, 4psubspset 34941 . . 3  |-  ( K  e.  D  ->  S  =  { x  |  ( x  C_  A  /\  A. p  e.  x  A. q  e.  x  A. r  e.  A  (
r  .<_  ( p  .\/  q )  ->  r  e.  x ) ) } )
65eleq2d 2537 . 2  |-  ( K  e.  D  ->  ( X  e.  S  <->  X  e.  { x  |  ( x 
C_  A  /\  A. p  e.  x  A. q  e.  x  A. r  e.  A  (
r  .<_  ( p  .\/  q )  ->  r  e.  x ) ) } ) )
7 fvex 5882 . . . . . 6  |-  ( Atoms `  K )  e.  _V
83, 7eqeltri 2551 . . . . 5  |-  A  e. 
_V
98ssex 4597 . . . 4  |-  ( X 
C_  A  ->  X  e.  _V )
109adantr 465 . . 3  |-  ( ( X  C_  A  /\  A. p  e.  X  A. q  e.  X  A. r  e.  A  (
r  .<_  ( p  .\/  q )  ->  r  e.  X ) )  ->  X  e.  _V )
11 sseq1 3530 . . . 4  |-  ( x  =  X  ->  (
x  C_  A  <->  X  C_  A
) )
12 eleq2 2540 . . . . . . . 8  |-  ( x  =  X  ->  (
r  e.  x  <->  r  e.  X ) )
1312imbi2d 316 . . . . . . 7  |-  ( x  =  X  ->  (
( r  .<_  ( p 
.\/  q )  -> 
r  e.  x )  <-> 
( r  .<_  ( p 
.\/  q )  -> 
r  e.  X ) ) )
1413ralbidv 2906 . . . . . 6  |-  ( x  =  X  ->  ( A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  x )  <->  A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  X ) ) )
1514raleqbi1dv 3071 . . . . 5  |-  ( x  =  X  ->  ( A. q  e.  x  A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  x )  <->  A. q  e.  X  A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  X ) ) )
1615raleqbi1dv 3071 . . . 4  |-  ( x  =  X  ->  ( A. p  e.  x  A. q  e.  x  A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  x )  <->  A. p  e.  X  A. q  e.  X  A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  X ) ) )
1711, 16anbi12d 710 . . 3  |-  ( x  =  X  ->  (
( x  C_  A  /\  A. p  e.  x  A. q  e.  x  A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  x ) )  <->  ( X  C_  A  /\  A. p  e.  X  A. q  e.  X  A. r  e.  A  ( r  .<_  ( p  .\/  q )  ->  r  e.  X
) ) ) )
1810, 17elab3 3262 . 2  |-  ( X  e.  { x  |  ( x  C_  A  /\  A. p  e.  x  A. q  e.  x  A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  x ) ) }  <->  ( X  C_  A  /\  A. p  e.  X  A. q  e.  X  A. r  e.  A  ( r  .<_  ( p  .\/  q
)  ->  r  e.  X ) ) )
196, 18syl6bb 261 1  |-  ( K  e.  D  ->  ( X  e.  S  <->  ( X  C_  A  /\  A. p  e.  X  A. q  e.  X  A. r  e.  A  ( r  .<_  ( p  .\/  q
)  ->  r  e.  X ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2817   _Vcvv 3118    C_ wss 3481   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   lecple 14579   joincjn 15448   Atomscatm 34461   PSubSpcpsubsp 34693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-psubsp 34700
This theorem is referenced by:  ispsubsp2  34943  0psubN  34946  snatpsubN  34947  linepsubN  34949  atpsubN  34950  psubssat  34951  pmapsub  34965  pclclN  35088  pclfinN  35097
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