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Theorem elpaddat 30601
Description: Membership in a projective subspace sum with a point. (Contributed by NM, 29-Jan-2012.)
Hypotheses
Ref Expression
paddfval.l  |-  .<_  =  ( le `  K )
paddfval.j  |-  .\/  =  ( join `  K )
paddfval.a  |-  A  =  ( Atoms `  K )
paddfval.p  |-  .+  =  ( + P `  K
)
Assertion
Ref Expression
elpaddat  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  -> 
( S  e.  ( X  .+  { Q } )  <->  ( S  e.  A  /\  E. p  e.  X  S  .<_  ( p  .\/  Q ) ) ) )
Distinct variable groups:    A, p    K, p    X, p    .\/ , p    .<_ , p    S, p    Q, p
Allowed substitution hint:    .+ ( p)

Proof of Theorem elpaddat
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 simpl1 960 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  K  e.  Lat )
2 simpl2 961 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  X  C_  A )
3 simpl3 962 . . . 4  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  Q  e.  A )
43snssd 3943 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  { Q }  C_  A
)
5 simpr 448 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  X  =/=  (/) )
6 snnzg 3921 . . . 4  |-  ( Q  e.  A  ->  { Q }  =/=  (/) )
73, 6syl 16 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  { Q }  =/=  (/) )
8 paddfval.l . . . 4  |-  .<_  =  ( le `  K )
9 paddfval.j . . . 4  |-  .\/  =  ( join `  K )
10 paddfval.a . . . 4  |-  A  =  ( Atoms `  K )
11 paddfval.p . . . 4  |-  .+  =  ( + P `  K
)
128, 9, 10, 11elpaddn0 30597 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  { Q }  C_  A
)  /\  ( X  =/=  (/)  /\  { Q }  =/=  (/) ) )  -> 
( S  e.  ( X  .+  { Q } )  <->  ( S  e.  A  /\  E. p  e.  X  E. r  e.  { Q } S  .<_  ( p  .\/  r
) ) ) )
131, 2, 4, 5, 7, 12syl32anc 1192 . 2  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  -> 
( S  e.  ( X  .+  { Q } )  <->  ( S  e.  A  /\  E. p  e.  X  E. r  e.  { Q } S  .<_  ( p  .\/  r
) ) ) )
14 oveq2 6089 . . . . . . 7  |-  ( r  =  Q  ->  (
p  .\/  r )  =  ( p  .\/  Q ) )
1514breq2d 4224 . . . . . 6  |-  ( r  =  Q  ->  ( S  .<_  ( p  .\/  r )  <->  S  .<_  ( p  .\/  Q ) ) )
1615rexsng 3847 . . . . 5  |-  ( Q  e.  A  ->  ( E. r  e.  { Q } S  .<_  ( p 
.\/  r )  <->  S  .<_  ( p  .\/  Q ) ) )
173, 16syl 16 . . . 4  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  -> 
( E. r  e. 
{ Q } S  .<_  ( p  .\/  r
)  <->  S  .<_  ( p 
.\/  Q ) ) )
1817rexbidv 2726 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  -> 
( E. p  e.  X  E. r  e. 
{ Q } S  .<_  ( p  .\/  r
)  <->  E. p  e.  X  S  .<_  ( p  .\/  Q ) ) )
1918anbi2d 685 . 2  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  -> 
( ( S  e.  A  /\  E. p  e.  X  E. r  e.  { Q } S  .<_  ( p  .\/  r
) )  <->  ( S  e.  A  /\  E. p  e.  X  S  .<_  ( p  .\/  Q ) ) ) )
2013, 19bitrd 245 1  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  -> 
( S  e.  ( X  .+  { Q } )  <->  ( S  e.  A  /\  E. p  e.  X  S  .<_  ( p  .\/  Q ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706    C_ wss 3320   (/)c0 3628   {csn 3814   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   lecple 13536   joincjn 14401   Latclat 14474   Atomscatm 30061   + Pcpadd 30592
This theorem is referenced by:  elpaddatiN  30602  elpadd2at  30603  pclfinclN  30747
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-lub 14431  df-join 14433  df-lat 14475  df-ats 30065  df-padd 30593
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