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Theorem ispsubsp 33747
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 33746 . . 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 2524 . 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 5812 . . . . . 6  |-  ( Atoms `  K )  e.  _V
83, 7eqeltri 2538 . . . . 5  |-  A  e. 
_V
98ssex 4547 . . . 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 3488 . . . 4  |-  ( x  =  X  ->  (
x  C_  A  <->  X  C_  A
) )
12 eleq2 2527 . . . . . . . 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 2846 . . . . . 6  |-  ( x  =  X  ->  ( A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  x )  <->  A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  X ) ) )
1514raleqbi1dv 3031 . . . . 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 3031 . . . 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 3220 . 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 1370    e. wcel 1758   {cab 2439   A.wral 2799   _Vcvv 3078    C_ wss 3439   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   lecple 14367   joincjn 15236   Atomscatm 33266   PSubSpcpsubsp 33498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-psubsp 33505
This theorem is referenced by:  ispsubsp2  33748  0psubN  33751  snatpsubN  33752  linepsubN  33754  atpsubN  33755  psubssat  33756  pmapsub  33770  pclclN  33893  pclfinN  33902
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