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Theorem elpadd2at 35927
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 994 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  K  e.  Lat )
2 simp2 995 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  Q  e.  A )
32snssd 4161 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  { Q }  C_  A )
4 simp3 996 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  R  e.  A )
5 snnzg 4133 . . . 4  |-  ( Q  e.  A  ->  { Q }  =/=  (/) )
653ad2ant2 1016 . . 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 35925 . . 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 1229 . 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 6277 . . . . . 6  |-  ( r  =  Q  ->  (
r  .\/  R )  =  ( Q  .\/  R ) )
1413breq2d 4451 . . . . 5  |-  ( r  =  Q  ->  ( S  .<_  ( r  .\/  R )  <->  S  .<_  ( Q 
.\/  R ) ) )
1514rexsng 4052 . . . 4  |-  ( Q  e.  A  ->  ( E. r  e.  { Q } S  .<_  ( r 
.\/  R )  <->  S  .<_  ( Q  .\/  R ) ) )
16153ad2ant2 1016 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  ( E. r  e. 
{ Q } S  .<_  ( r  .\/  R
)  <->  S  .<_  ( Q 
.\/  R ) ) )
1716anbi2d 701 . 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 253 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 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805    C_ wss 3461   (/)c0 3783   {csn 4016   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   lecple 14791   joincjn 15772   Latclat 15874   Atomscatm 35385   +Pcpadd 35916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-lub 15803  df-join 15805  df-lat 15875  df-ats 35389  df-padd 35917
This theorem is referenced by:  elpadd2at2  35928
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