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Theorem paddvaln0N 30059
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 30058 . . 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 4107 . . . . 5  |-  ( p  =  s  ->  (
p  .<_  ( q  .\/  r )  <->  s  .<_  ( q  .\/  r ) ) )
762rexbidv 2662 . . . 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 2999 . . 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 254 . 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 2356 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620   {crab 2623    C_ wss 3228   (/)c0 3531   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   lecple 13312   joincjn 14177   Latclat 14250   Atomscatm 29522   + Pcpadd 30053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-undef 6385  df-riota 6391  df-lub 14207  df-join 14209  df-lat 14251  df-ats 29526  df-padd 30054
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