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Theorem paddvaln0N 30287
Description: Projective subspace sum operation value for non-empty sets. (Contributed by NM, 27-Jan-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
paddvaln0N  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  -> 
( X  .+  Y
)  =  { p  e.  A  |  E. q  e.  X  E. r  e.  Y  p  .<_  ( q  .\/  r
) } )
Distinct variable groups:    A, p, q, r    K, p    r,
q, K    X, p, q    Y, p, q, r    .\/ , p    .<_ , p    A, q, r    .\/ , q, r    .<_ , q, r    X, r
Allowed substitution hints:    .+ ( r, q, p)

Proof of Theorem paddvaln0N
Dummy variable  s is distinct from all other variables.
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, 4elpaddn0 30286 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  -> 
( s  e.  ( X  .+  Y )  <-> 
( s  e.  A  /\  E. q  e.  X  E. r  e.  Y  s  .<_  ( q  .\/  r ) ) ) )
6 breq1 4179 . . . . 5  |-  ( p  =  s  ->  (
p  .<_  ( q  .\/  r )  <->  s  .<_  ( q  .\/  r ) ) )
762rexbidv 2713 . . . 4  |-  ( p  =  s  ->  ( E. q  e.  X  E. r  e.  Y  p  .<_  ( q  .\/  r )  <->  E. q  e.  X  E. r  e.  Y  s  .<_  ( q  .\/  r ) ) )
87elrab 3056 . . 3  |-  ( s  e.  { p  e.  A  |  E. q  e.  X  E. r  e.  Y  p  .<_  ( q  .\/  r ) }  <->  ( s  e.  A  /\  E. q  e.  X  E. r  e.  Y  s  .<_  ( q  .\/  r ) ) )
95, 8syl6bbr 255 . 2  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  -> 
( s  e.  ( X  .+  Y )  <-> 
s  e.  { p  e.  A  |  E. q  e.  X  E. r  e.  Y  p  .<_  ( q  .\/  r
) } ) )
109eqrdv 2406 1  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  -> 
( X  .+  Y
)  =  { p  e.  A  |  E. q  e.  X  E. r  e.  Y  p  .<_  ( q  .\/  r
) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   E.wrex 2671   {crab 2674    C_ wss 3284   (/)c0 3592   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   lecple 13495   joincjn 14360   Latclat 14433   Atomscatm 29750   + Pcpadd 30281
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-lub 14390  df-join 14392  df-lat 14434  df-ats 29754  df-padd 30282
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