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Theorem elpaddat 33077
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 1008 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  K  e.  Lat )
2 simpl2 1009 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  X  C_  A )
3 simpl3 1010 . . . 4  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  Q  e.  A )
43snssd 4148 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  { Q }  C_  A
)
5 simpr 462 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  X  =/=  (/) )
6 snnzg 4120 . . . 4  |-  ( Q  e.  A  ->  { Q }  =/=  (/) )
73, 6syl 17 . . 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 33073 . . 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 1272 . 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 6313 . . . . . . 7  |-  ( r  =  Q  ->  (
p  .\/  r )  =  ( p  .\/  Q ) )
1514breq2d 4438 . . . . . 6  |-  ( r  =  Q  ->  ( S  .<_  ( p  .\/  r )  <->  S  .<_  ( p  .\/  Q ) ) )
1615rexsng 4038 . . . . 5  |-  ( Q  e.  A  ->  ( E. r  e.  { Q } S  .<_  ( p 
.\/  r )  <->  S  .<_  ( p  .\/  Q ) ) )
173, 16syl 17 . . . 4  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  -> 
( E. r  e. 
{ Q } S  .<_  ( p  .\/  r
)  <->  S  .<_  ( p 
.\/  Q ) ) )
1817rexbidv 2946 . . 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 708 . 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 256 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783    C_ wss 3442   (/)c0 3767   {csn 4002   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   lecple 15159   joincjn 16140   Latclat 16242   Atomscatm 32537   +Pcpadd 33068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-lub 16171  df-join 16173  df-lat 16243  df-ats 32541  df-padd 33069
This theorem is referenced by:  elpaddatiN  33078  elpadd2at  33079  pclfinclN  33223
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