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Theorem paddfval 30608
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 2809 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 paddfval.p . . 3  |-  .+  =  ( + P `  K
)
3 fveq2 5541 . . . . . . 7  |-  ( h  =  K  ->  ( Atoms `  h )  =  ( Atoms `  K )
)
4 paddfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2346 . . . . . 6  |-  ( h  =  K  ->  ( Atoms `  h )  =  A )
65pweqd 3643 . . . . 5  |-  ( h  =  K  ->  ~P ( Atoms `  h )  =  ~P A )
7 eqidd 2297 . . . . . . . . 9  |-  ( h  =  K  ->  p  =  p )
8 fveq2 5541 . . . . . . . . . 10  |-  ( h  =  K  ->  ( le `  h )  =  ( le `  K
) )
9 paddfval.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2346 . . . . . . . . 9  |-  ( h  =  K  ->  ( le `  h )  = 
.<_  )
11 fveq2 5541 . . . . . . . . . . 11  |-  ( h  =  K  ->  ( join `  h )  =  ( join `  K
) )
12 paddfval.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
1311, 12syl6eqr 2346 . . . . . . . . . 10  |-  ( h  =  K  ->  ( join `  h )  = 
.\/  )
1413oveqd 5891 . . . . . . . . 9  |-  ( h  =  K  ->  (
q ( join `  h
) r )  =  ( q  .\/  r
) )
157, 10, 14breq123d 4053 . . . . . . . 8  |-  ( h  =  K  ->  (
p ( le `  h ) ( q ( join `  h
) r )  <->  p  .<_  ( q  .\/  r ) ) )
16152rexbidv 2599 . . . . . . 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 2796 . . . . . 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 3342 . . . . 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 5926 . . . 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 30607 . . . 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 5555 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
224, 21eqeltri 2366 . . . . . 6  |-  A  e. 
_V
2322pwex 4209 . . . . 5  |-  ~P A  e.  _V
2423, 23mpt2ex 6214 . . . 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 5618 . . 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 2340 . 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 1632    e. wcel 1696   E.wrex 2557   {crab 2560   _Vcvv 2801    u. cun 3163   ~Pcpw 3638   class class class wbr 4039   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   lecple 13231   joincjn 14094   Atomscatm 30075   + Pcpadd 30606
This theorem is referenced by:  paddval  30609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-padd 30607
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