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Theorem paddfval 29912
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 2908 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 paddfval.p . . 3  |-  .+  =  ( + P `  K
)
3 fveq2 5669 . . . . . . 7  |-  ( h  =  K  ->  ( Atoms `  h )  =  ( Atoms `  K )
)
4 paddfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2438 . . . . . 6  |-  ( h  =  K  ->  ( Atoms `  h )  =  A )
65pweqd 3748 . . . . 5  |-  ( h  =  K  ->  ~P ( Atoms `  h )  =  ~P A )
7 eqidd 2389 . . . . . . . . 9  |-  ( h  =  K  ->  p  =  p )
8 fveq2 5669 . . . . . . . . . 10  |-  ( h  =  K  ->  ( le `  h )  =  ( le `  K
) )
9 paddfval.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2438 . . . . . . . . 9  |-  ( h  =  K  ->  ( le `  h )  = 
.<_  )
11 fveq2 5669 . . . . . . . . . . 11  |-  ( h  =  K  ->  ( join `  h )  =  ( join `  K
) )
12 paddfval.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
1311, 12syl6eqr 2438 . . . . . . . . . 10  |-  ( h  =  K  ->  ( join `  h )  = 
.\/  )
1413oveqd 6038 . . . . . . . . 9  |-  ( h  =  K  ->  (
q ( join `  h
) r )  =  ( q  .\/  r
) )
157, 10, 14breq123d 4168 . . . . . . . 8  |-  ( h  =  K  ->  (
p ( le `  h ) ( q ( join `  h
) r )  <->  p  .<_  ( q  .\/  r ) ) )
16152rexbidv 2693 . . . . . . 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 2895 . . . . . 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 3445 . . . . 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 6076 . . . 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 29911 . . . 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 5683 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
224, 21eqeltri 2458 . . . . . 6  |-  A  e. 
_V
2322pwex 4324 . . . . 5  |-  ~P A  e.  _V
2423, 23mpt2ex 6365 . . . 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 5746 . . 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 2432 . 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 16 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 1649    e. wcel 1717   E.wrex 2651   {crab 2654   _Vcvv 2900    u. cun 3262   ~Pcpw 3743   class class class wbr 4154   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   lecple 13464   joincjn 14329   Atomscatm 29379   + Pcpadd 29910
This theorem is referenced by:  paddval  29913
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-padd 29911
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