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Theorem elpaddat 30615
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 958 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  K  e.  Lat )
2 simpl2 959 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  X  C_  A )
3 simpl3 960 . . . 4  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  Q  e.  A )
43snssd 3776 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  { Q }  C_  A
)
5 simpr 447 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  X  =/=  (/) )
6 snnzg 3756 . . . 4  |-  ( Q  e.  A  ->  { Q }  =/=  (/) )
73, 6syl 15 . . 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 30611 . . 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 1190 . 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 5882 . . . . . . 7  |-  ( r  =  Q  ->  (
p  .\/  r )  =  ( p  .\/  Q ) )
1514breq2d 4051 . . . . . 6  |-  ( r  =  Q  ->  ( S  .<_  ( p  .\/  r )  <->  S  .<_  ( p  .\/  Q ) ) )
1615rexsng 3686 . . . . 5  |-  ( Q  e.  A  ->  ( E. r  e.  { Q } S  .<_  ( p 
.\/  r )  <->  S  .<_  ( p  .\/  Q ) ) )
173, 16syl 15 . . . 4  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  -> 
( E. r  e. 
{ Q } S  .<_  ( p  .\/  r
)  <->  S  .<_  ( p 
.\/  Q ) ) )
1817rexbidv 2577 . . 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 684 . 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 244 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    C_ wss 3165   (/)c0 3468   {csn 3653   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   Latclat 14167   Atomscatm 30075   + Pcpadd 30606
This theorem is referenced by:  elpaddatiN  30616  elpadd2at  30617  pclfinclN  30761
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  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-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-lub 14124  df-join 14126  df-lat 14168  df-ats 30079  df-padd 30607
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