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Theorem paddvaln0N 30672
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 30671 . . 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 4218 . . . . 5  |-  ( p  =  s  ->  (
p  .<_  ( q  .\/  r )  <->  s  .<_  ( q  .\/  r ) ) )
762rexbidv 2750 . . . 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 3094 . . 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 256 . 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 2436 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   {crab 2711    C_ wss 3322   (/)c0 3630   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   lecple 13541   joincjn 14406   Latclat 14479   Atomscatm 30135   + Pcpadd 30666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-lub 14436  df-join 14438  df-lat 14480  df-ats 30139  df-padd 30667
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