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Theorem ispsubsp 30544
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 30543 . . 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 2505 . 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 5744 . . . . . 6  |-  ( Atoms `  K )  e.  _V
83, 7eqeltri 2508 . . . . 5  |-  A  e. 
_V
98ssex 4349 . . . 4  |-  ( X 
C_  A  ->  X  e.  _V )
109adantr 453 . . 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 3371 . . . 4  |-  ( x  =  X  ->  (
x  C_  A  <->  X  C_  A
) )
12 eleq2 2499 . . . . . . . 8  |-  ( x  =  X  ->  (
r  e.  x  <->  r  e.  X ) )
1312imbi2d 309 . . . . . . 7  |-  ( x  =  X  ->  (
( r  .<_  ( p 
.\/  q )  -> 
r  e.  x )  <-> 
( r  .<_  ( p 
.\/  q )  -> 
r  e.  X ) ) )
1413ralbidv 2727 . . . . . 6  |-  ( x  =  X  ->  ( A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  x )  <->  A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  X ) ) )
1514raleqbi1dv 2914 . . . . 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 2914 . . . 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 693 . . 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 3091 . 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 254 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   _Vcvv 2958    C_ wss 3322   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   lecple 13538   joincjn 14403   Atomscatm 30063   PSubSpcpsubsp 30295
This theorem is referenced by:  ispsubsp2  30545  0psubN  30548  snatpsubN  30549  linepsubN  30551  atpsubN  30552  psubssat  30553  pmapsub  30567  pclclN  30690  pclfinN  30699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-psubsp 30302
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