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Theorem paddfval 35934
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 3043 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 paddfval.p . . 3  |-  .+  =  ( +P `  K
)
3 fveq2 5774 . . . . . . 7  |-  ( h  =  K  ->  ( Atoms `  h )  =  ( Atoms `  K )
)
4 paddfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2441 . . . . . 6  |-  ( h  =  K  ->  ( Atoms `  h )  =  A )
65pweqd 3932 . . . . 5  |-  ( h  =  K  ->  ~P ( Atoms `  h )  =  ~P A )
7 eqidd 2383 . . . . . . . . 9  |-  ( h  =  K  ->  p  =  p )
8 fveq2 5774 . . . . . . . . . 10  |-  ( h  =  K  ->  ( le `  h )  =  ( le `  K
) )
9 paddfval.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2441 . . . . . . . . 9  |-  ( h  =  K  ->  ( le `  h )  = 
.<_  )
11 fveq2 5774 . . . . . . . . . . 11  |-  ( h  =  K  ->  ( join `  h )  =  ( join `  K
) )
12 paddfval.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
1311, 12syl6eqr 2441 . . . . . . . . . 10  |-  ( h  =  K  ->  ( join `  h )  = 
.\/  )
1413oveqd 6213 . . . . . . . . 9  |-  ( h  =  K  ->  (
q ( join `  h
) r )  =  ( q  .\/  r
) )
157, 10, 14breq123d 4381 . . . . . . . 8  |-  ( h  =  K  ->  (
p ( le `  h ) ( q ( join `  h
) r )  <->  p  .<_  ( q  .\/  r ) ) )
16152rexbidv 2900 . . . . . . 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 3029 . . . . . 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 3572 . . . . 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 6258 . . . 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 35933 . . . 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 5784 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
224, 21eqeltri 2466 . . . . . 6  |-  A  e. 
_V
2322pwex 4548 . . . . 5  |-  ~P A  e.  _V
2423, 23mpt2ex 6776 . . . 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 5857 . . 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 2435 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826   E.wrex 2733   {crab 2736   _Vcvv 3034    u. cun 3387   ~Pcpw 3927   class class class wbr 4367   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198   lecple 14709   joincjn 15690   Atomscatm 35401   +Pcpadd 35932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-padd 35933
This theorem is referenced by:  paddval  35935
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