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Theorem ispsubsp 29934
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 29933 . . 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 2350 . 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 5539 . . . . . 6  |-  ( Atoms `  K )  e.  _V
83, 7eqeltri 2353 . . . . 5  |-  A  e. 
_V
98ssex 4158 . . . 4  |-  ( X 
C_  A  ->  X  e.  _V )
109adantr 451 . . 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 3199 . . . 4  |-  ( x  =  X  ->  (
x  C_  A  <->  X  C_  A
) )
12 eleq2 2344 . . . . . . . 8  |-  ( x  =  X  ->  (
r  e.  x  <->  r  e.  X ) )
1312imbi2d 307 . . . . . . 7  |-  ( x  =  X  ->  (
( r  .<_  ( p 
.\/  q )  -> 
r  e.  x )  <-> 
( r  .<_  ( p 
.\/  q )  -> 
r  e.  X ) ) )
1413ralbidv 2563 . . . . . 6  |-  ( x  =  X  ->  ( A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  x )  <->  A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  X ) ) )
1514raleqbi1dv 2744 . . . . 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 2744 . . . 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 691 . . 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 2921 . 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 252 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   _Vcvv 2788    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453   PSubSpcpsubsp 29685
This theorem is referenced by:  ispsubsp2  29935  0psubN  29938  snatpsubN  29939  linepsubN  29941  atpsubN  29942  psubssat  29943  pmapsub  29957  pclclN  30080  pclfinN  30089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-psubsp 29692
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