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Theorem elpadd2at 30300
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 957 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  K  e.  Lat )
2 simp2 958 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  Q  e.  A )
32snssd 3911 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  { Q }  C_  A )
4 simp3 959 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  R  e.  A )
5 snnzg 3889 . . . 4  |-  ( Q  e.  A  ->  { Q }  =/=  (/) )
653ad2ant2 979 . . 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 30298 . . 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 1187 . 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 6055 . . . . . 6  |-  ( r  =  Q  ->  (
r  .\/  R )  =  ( Q  .\/  R ) )
1413breq2d 4192 . . . . 5  |-  ( r  =  Q  ->  ( S  .<_  ( r  .\/  R )  <->  S  .<_  ( Q 
.\/  R ) ) )
1514rexsng 3815 . . . 4  |-  ( Q  e.  A  ->  ( E. r  e.  { Q } S  .<_  ( r 
.\/  R )  <->  S  .<_  ( Q  .\/  R ) ) )
16153ad2ant2 979 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  A  /\  R  e.  A )  ->  ( E. r  e. 
{ Q } S  .<_  ( r  .\/  R
)  <->  S  .<_  ( Q 
.\/  R ) ) )
1716anbi2d 685 . 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 245 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675    C_ wss 3288   (/)c0 3596   {csn 3782   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   lecple 13499   joincjn 14364   Latclat 14437   Atomscatm 29758   + Pcpadd 30289
This theorem is referenced by:  elpadd2at2  30301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-lub 14394  df-join 14396  df-lat 14438  df-ats 29762  df-padd 30290
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