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Theorem elpaddat 34618
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 999 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  K  e.  Lat )
2 simpl2 1000 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  X  C_  A )
3 simpl3 1001 . . . 4  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  Q  e.  A )
43snssd 4172 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  { Q }  C_  A
)
5 simpr 461 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  X  =/=  (/) )
6 snnzg 4144 . . . 4  |-  ( Q  e.  A  ->  { Q }  =/=  (/) )
73, 6syl 16 . . 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 34614 . . 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 1236 . 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 6292 . . . . . . 7  |-  ( r  =  Q  ->  (
p  .\/  r )  =  ( p  .\/  Q ) )
1514breq2d 4459 . . . . . 6  |-  ( r  =  Q  ->  ( S  .<_  ( p  .\/  r )  <->  S  .<_  ( p  .\/  Q ) ) )
1615rexsng 4063 . . . . 5  |-  ( Q  e.  A  ->  ( E. r  e.  { Q } S  .<_  ( p 
.\/  r )  <->  S  .<_  ( p  .\/  Q ) ) )
173, 16syl 16 . . . 4  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  -> 
( E. r  e. 
{ Q } S  .<_  ( p  .\/  r
)  <->  S  .<_  ( p 
.\/  Q ) ) )
1817rexbidv 2973 . . 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 703 . 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 253 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    C_ wss 3476   (/)c0 3785   {csn 4027   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   lecple 14562   joincjn 15431   Latclat 15532   Atomscatm 34078   +Pcpadd 34609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-lub 15461  df-join 15463  df-lat 15533  df-ats 34082  df-padd 34610
This theorem is referenced by:  elpaddatiN  34619  elpadd2at  34620  pclfinclN  34764
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