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Theorem paddfval 29986
Description: Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.)
Hypotheses
Ref Expression
paddfval.l  |-  .<_  =  ( le `  K )
paddfval.j  |-  .\/  =  ( join `  K )
paddfval.a  |-  A  =  ( Atoms `  K )
paddfval.p  |-  .+  =  ( + P `  K
)
Assertion
Ref Expression
paddfval  |-  ( K  e.  B  ->  .+  =  ( m  e.  ~P A ,  n  e.  ~P A  |->  ( ( m  u.  n )  u.  { p  e.  A  |  E. q  e.  m  E. r  e.  n  p  .<_  ( q  .\/  r ) } ) ) )
Distinct variable groups:    m, n, p, A    m, q, r, K, n, p
Allowed substitution hints:    A( r, q)    B( m, n, r, q, p)    .+ ( m, n, r, q, p)    .\/ ( m, n, r, q, p)    .<_ ( m, n, r, q, p)

Proof of Theorem paddfval
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 paddfval.p . . 3  |-  .+  =  ( + P `  K
)
3 fveq2 5525 . . . . . . 7  |-  ( h  =  K  ->  ( Atoms `  h )  =  ( Atoms `  K )
)
4 paddfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2333 . . . . . 6  |-  ( h  =  K  ->  ( Atoms `  h )  =  A )
65pweqd 3630 . . . . 5  |-  ( h  =  K  ->  ~P ( Atoms `  h )  =  ~P A )
7 eqidd 2284 . . . . . . . . 9  |-  ( h  =  K  ->  p  =  p )
8 fveq2 5525 . . . . . . . . . 10  |-  ( h  =  K  ->  ( le `  h )  =  ( le `  K
) )
9 paddfval.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2333 . . . . . . . . 9  |-  ( h  =  K  ->  ( le `  h )  = 
.<_  )
11 fveq2 5525 . . . . . . . . . . 11  |-  ( h  =  K  ->  ( join `  h )  =  ( join `  K
) )
12 paddfval.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
1311, 12syl6eqr 2333 . . . . . . . . . 10  |-  ( h  =  K  ->  ( join `  h )  = 
.\/  )
1413oveqd 5875 . . . . . . . . 9  |-  ( h  =  K  ->  (
q ( join `  h
) r )  =  ( q  .\/  r
) )
157, 10, 14breq123d 4037 . . . . . . . 8  |-  ( h  =  K  ->  (
p ( le `  h ) ( q ( join `  h
) r )  <->  p  .<_  ( q  .\/  r ) ) )
16152rexbidv 2586 . . . . . . 7  |-  ( h  =  K  ->  ( E. q  e.  m  E. r  e.  n  p ( le `  h ) ( q ( join `  h
) r )  <->  E. q  e.  m  E. r  e.  n  p  .<_  ( q  .\/  r ) ) )
175, 16rabeqbidv 2783 . . . . . 6  |-  ( h  =  K  ->  { p  e.  ( Atoms `  h )  |  E. q  e.  m  E. r  e.  n  p ( le `  h ) ( q ( join `  h
) r ) }  =  { p  e.  A  |  E. q  e.  m  E. r  e.  n  p  .<_  ( q  .\/  r ) } )
1817uneq2d 3329 . . . . 5  |-  ( h  =  K  ->  (
( m  u.  n
)  u.  { p  e.  ( Atoms `  h )  |  E. q  e.  m  E. r  e.  n  p ( le `  h ) ( q ( join `  h
) r ) } )  =  ( ( m  u.  n )  u.  { p  e.  A  |  E. q  e.  m  E. r  e.  n  p  .<_  ( q  .\/  r ) } ) )
196, 6, 18mpt2eq123dv 5910 . . . 4  |-  ( h  =  K  ->  (
m  e.  ~P ( Atoms `  h ) ,  n  e.  ~P ( Atoms `  h )  |->  ( ( m  u.  n
)  u.  { p  e.  ( Atoms `  h )  |  E. q  e.  m  E. r  e.  n  p ( le `  h ) ( q ( join `  h
) r ) } ) )  =  ( m  e.  ~P A ,  n  e.  ~P A  |->  ( ( m  u.  n )  u. 
{ p  e.  A  |  E. q  e.  m  E. r  e.  n  p  .<_  ( q  .\/  r ) } ) ) )
20 df-padd 29985 . . . 4  |-  + P  =  ( h  e. 
_V  |->  ( m  e. 
~P ( Atoms `  h
) ,  n  e. 
~P ( Atoms `  h
)  |->  ( ( m  u.  n )  u. 
{ p  e.  (
Atoms `  h )  |  E. q  e.  m  E. r  e.  n  p ( le `  h ) ( q ( join `  h
) r ) } ) ) )
21 fvex 5539 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
224, 21eqeltri 2353 . . . . . 6  |-  A  e. 
_V
2322pwex 4193 . . . . 5  |-  ~P A  e.  _V
2423, 23mpt2ex 6198 . . . 4  |-  ( m  e.  ~P A ,  n  e.  ~P A  |->  ( ( m  u.  n )  u.  {
p  e.  A  |  E. q  e.  m  E. r  e.  n  p  .<_  ( q  .\/  r ) } ) )  e.  _V
2519, 20, 24fvmpt 5602 . . 3  |-  ( K  e.  _V  ->  ( + P `  K )  =  ( m  e. 
~P A ,  n  e.  ~P A  |->  ( ( m  u.  n )  u.  { p  e.  A  |  E. q  e.  m  E. r  e.  n  p  .<_  ( q  .\/  r ) } ) ) )
262, 25syl5eq 2327 . 2  |-  ( K  e.  _V  ->  .+  =  ( m  e.  ~P A ,  n  e.  ~P A  |->  ( ( m  u.  n )  u.  { p  e.  A  |  E. q  e.  m  E. r  e.  n  p  .<_  ( q  .\/  r ) } ) ) )
271, 26syl 15 1  |-  ( K  e.  B  ->  .+  =  ( m  e.  ~P A ,  n  e.  ~P A  |->  ( ( m  u.  n )  u.  { p  e.  A  |  E. q  e.  m  E. r  e.  n  p  .<_  ( q  .\/  r ) } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788    u. cun 3150   ~Pcpw 3625   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   lecple 13215   joincjn 14078   Atomscatm 29453   + Pcpadd 29984
This theorem is referenced by:  paddval  29987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-padd 29985
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