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Theorem ispsubsp 30556
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 30555 . . 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 2363 . 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 5555 . . . . . 6  |-  ( Atoms `  K )  e.  _V
83, 7eqeltri 2366 . . . . 5  |-  A  e. 
_V
98ssex 4174 . . . 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 3212 . . . 4  |-  ( x  =  X  ->  (
x  C_  A  <->  X  C_  A
) )
12 eleq2 2357 . . . . . . . 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 2576 . . . . . 6  |-  ( x  =  X  ->  ( A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  x )  <->  A. r  e.  A  ( r  .<_  ( p 
.\/  q )  -> 
r  e.  X ) ) )
1514raleqbi1dv 2757 . . . . 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 2757 . . . 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 2934 . 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 1632    e. wcel 1696   {cab 2282   A.wral 2556   _Vcvv 2801    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   Atomscatm 30075   PSubSpcpsubsp 30307
This theorem is referenced by:  ispsubsp2  30557  0psubN  30560  snatpsubN  30561  linepsubN  30563  atpsubN  30564  psubssat  30565  pmapsub  30579  pclclN  30702  pclfinN  30711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-psubsp 30314
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