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Theorem elpadd2at 30701
Description: Membership in a projective subspace sum of two points. (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
elpadd2at  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  ( S  e.  ( { Q }  .+  { R } )  <->  ( S  e.  A  /\  S  .<_  ( Q  .\/  R ) ) ) )

Proof of Theorem elpadd2at
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 simp1 958 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  K  e.  Lat )
2 simp2 959 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  Q  e.  A )
32snssd 3967 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  { Q }  C_  A )
4 simp3 960 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  R  e.  A )
5 snnzg 3945 . . . 4  |-  ( Q  e.  A  ->  { Q }  =/=  (/) )
653ad2ant2 980 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  { Q }  =/=  (/) )
7 paddfval.l . . . 4  |-  .<_  =  ( le `  K )
8 paddfval.j . . . 4  |-  .\/  =  ( join `  K )
9 paddfval.a . . . 4  |-  A  =  ( Atoms `  K )
10 paddfval.p . . . 4  |-  .+  =  ( + P `  K
)
117, 8, 9, 10elpaddat 30699 . . 3  |-  ( ( ( K  e.  Lat  /\ 
{ Q }  C_  A  /\  R  e.  A
)  /\  { Q }  =/=  (/) )  ->  ( S  e.  ( { Q }  .+  { R } )  <->  ( S  e.  A  /\  E. r  e.  { Q } S  .<_  ( r  .\/  R
) ) ) )
121, 3, 4, 6, 11syl31anc 1188 . 2  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  ( S  e.  ( { Q }  .+  { R } )  <->  ( S  e.  A  /\  E. r  e.  { Q } S  .<_  ( r  .\/  R
) ) ) )
13 oveq1 6117 . . . . . 6  |-  ( r  =  Q  ->  (
r  .\/  R )  =  ( Q  .\/  R ) )
1413breq2d 4249 . . . . 5  |-  ( r  =  Q  ->  ( S  .<_  ( r  .\/  R )  <->  S  .<_  ( Q 
.\/  R ) ) )
1514rexsng 3871 . . . 4  |-  ( Q  e.  A  ->  ( E. r  e.  { Q } S  .<_  ( r 
.\/  R )  <->  S  .<_  ( Q  .\/  R ) ) )
16153ad2ant2 980 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  ( E. r  e. 
{ Q } S  .<_  ( r  .\/  R
)  <->  S  .<_  ( Q 
.\/  R ) ) )
1716anbi2d 686 . 2  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  ( ( S  e.  A  /\  E. r  e.  { Q } S  .<_  ( r  .\/  R
) )  <->  ( S  e.  A  /\  S  .<_  ( Q  .\/  R ) ) ) )
1812, 17bitrd 246 1  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  ( S  e.  ( { Q }  .+  { R } )  <->  ( S  e.  A  /\  S  .<_  ( Q  .\/  R ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   E.wrex 2712    C_ wss 3306   (/)c0 3613   {csn 3838   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   lecple 13567   joincjn 14432   Latclat 14505   Atomscatm 30159   + Pcpadd 30690
This theorem is referenced by:  elpadd2at2  30702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-lub 14462  df-join 14464  df-lat 14506  df-ats 30163  df-padd 30691
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