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Theorem elpadd2at 33372
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 1009 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  K  e.  Lat )
2 simp2 1010 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  Q  e.  A )
32snssd 4085 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  { Q }  C_  A )
4 simp3 1011 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  R  e.  A )
5 snnzg 4057 . . . 4  |-  ( Q  e.  A  ->  { Q }  =/=  (/) )
653ad2ant2 1031 . . 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 33370 . . 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 1274 . 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 6282 . . . . . 6  |-  ( r  =  Q  ->  (
r  .\/  R )  =  ( Q  .\/  R ) )
1413breq2d 4385 . . . . 5  |-  ( r  =  Q  ->  ( S  .<_  ( r  .\/  R )  <->  S  .<_  ( Q 
.\/  R ) ) )
1514rexsng 3974 . . . 4  |-  ( Q  e.  A  ->  ( E. r  e.  { Q } S  .<_  ( r 
.\/  R )  <->  S  .<_  ( Q  .\/  R ) ) )
16153ad2ant2 1031 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  ( E. r  e. 
{ Q } S  .<_  ( r  .\/  R
)  <->  S  .<_  ( Q 
.\/  R ) ) )
1716anbi2d 715 . 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 261 1  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  ( S  e.  ( { Q }  .+  { R } )  <->  ( S  e.  A  /\  S  .<_  ( Q  .\/  R ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 986    = wceq 1447    e. wcel 1890    =/= wne 2621   E.wrex 2737    C_ wss 3371   (/)c0 3698   {csn 3935   class class class wbr 4373   ` cfv 5560  (class class class)co 6275   lecple 15207   joincjn 16199   Latclat 16301   Atomscatm 32830   +Pcpadd 33361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-rep 4486  ax-sep 4496  ax-nul 4505  ax-pow 4553  ax-pr 4611  ax-un 6570
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-pw 3920  df-sn 3936  df-pr 3938  df-op 3942  df-uni 4168  df-iun 4249  df-br 4374  df-opab 4433  df-mpt 4434  df-id 4726  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5524  df-fun 5562  df-fn 5563  df-f 5564  df-f1 5565  df-fo 5566  df-f1o 5567  df-fv 5568  df-riota 6237  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6780  df-2nd 6781  df-lub 16230  df-join 16232  df-lat 16302  df-ats 32834  df-padd 33362
This theorem is referenced by:  elpadd2at2  33373
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