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Theorem elpaddatiN 29921
Description: Consequence of membership in a projective subspace sum with a point. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
paddfval.l  |-  .<_  =  ( le `  K )
paddfval.j  |-  .\/  =  ( join `  K )
paddfval.a  |-  A  =  ( Atoms `  K )
paddfval.p  |-  .+  =  ( + P `  K
)
Assertion
Ref Expression
elpaddatiN  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  ( X  =/=  (/)  /\  R  e.  ( X  .+  { Q } ) ) )  ->  E. p  e.  X  R  .<_  ( p  .\/  Q ) )
Distinct variable groups:    A, p    K, p    X, p    .\/ , p    .<_ , p    Q, p    R, p
Allowed substitution hint:    .+ ( p)

Proof of Theorem elpaddatiN
StepHypRef Expression
1 paddfval.l . . . 4  |-  .<_  =  ( le `  K )
2 paddfval.j . . . 4  |-  .\/  =  ( join `  K )
3 paddfval.a . . . 4  |-  A  =  ( Atoms `  K )
4 paddfval.p . . . 4  |-  .+  =  ( + P `  K
)
51, 2, 3, 4elpaddat 29920 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  -> 
( R  e.  ( X  .+  { Q } )  <->  ( R  e.  A  /\  E. p  e.  X  R  .<_  ( p  .\/  Q ) ) ) )
6 simpr 448 . . 3  |-  ( ( R  e.  A  /\  E. p  e.  X  R  .<_  ( p  .\/  Q
) )  ->  E. p  e.  X  R  .<_  ( p  .\/  Q ) )
75, 6syl6bi 220 . 2  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  -> 
( R  e.  ( X  .+  { Q } )  ->  E. p  e.  X  R  .<_  ( p  .\/  Q ) ) )
87impr 603 1  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  ( X  =/=  (/)  /\  R  e.  ( X  .+  { Q } ) ) )  ->  E. p  e.  X  R  .<_  ( p  .\/  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   E.wrex 2652    C_ wss 3265   (/)c0 3573   {csn 3759   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   lecple 13465   joincjn 14330   Latclat 14403   Atomscatm 29380   + Pcpadd 29911
This theorem is referenced by:  osumcllem7N  30078  pexmidlem4N  30089
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-lub 14360  df-join 14362  df-lat 14404  df-ats 29384  df-padd 29912
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