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Theorem elpaddatiN 30539
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 30538 . . 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 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   Latclat 14466   Atomscatm 29998   + Pcpadd 30529
This theorem is referenced by:  osumcllem7N  30696  pexmidlem4N  30707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-lub 14423  df-join 14425  df-lat 14467  df-ats 30002  df-padd 30530
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