Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  paddfval Structured version   Unicode version

Theorem paddfval 33464
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 3000 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 paddfval.p . . 3  |-  .+  =  ( +P `  K
)
3 fveq2 5710 . . . . . . 7  |-  ( h  =  K  ->  ( Atoms `  h )  =  ( Atoms `  K )
)
4 paddfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2493 . . . . . 6  |-  ( h  =  K  ->  ( Atoms `  h )  =  A )
65pweqd 3884 . . . . 5  |-  ( h  =  K  ->  ~P ( Atoms `  h )  =  ~P A )
7 eqidd 2444 . . . . . . . . 9  |-  ( h  =  K  ->  p  =  p )
8 fveq2 5710 . . . . . . . . . 10  |-  ( h  =  K  ->  ( le `  h )  =  ( le `  K
) )
9 paddfval.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2493 . . . . . . . . 9  |-  ( h  =  K  ->  ( le `  h )  = 
.<_  )
11 fveq2 5710 . . . . . . . . . . 11  |-  ( h  =  K  ->  ( join `  h )  =  ( join `  K
) )
12 paddfval.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
1311, 12syl6eqr 2493 . . . . . . . . . 10  |-  ( h  =  K  ->  ( join `  h )  = 
.\/  )
1413oveqd 6127 . . . . . . . . 9  |-  ( h  =  K  ->  (
q ( join `  h
) r )  =  ( q  .\/  r
) )
157, 10, 14breq123d 4325 . . . . . . . 8  |-  ( h  =  K  ->  (
p ( le `  h ) ( q ( join `  h
) r )  <->  p  .<_  ( q  .\/  r ) ) )
16152rexbidv 2777 . . . . . . 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 2986 . . . . . 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 3529 . . . . 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 6167 . . . 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 33463 . . . 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 5720 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
224, 21eqeltri 2513 . . . . . 6  |-  A  e. 
_V
2322pwex 4494 . . . . 5  |-  ~P A  e.  _V
2423, 23mpt2ex 6669 . . . 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 5793 . . 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 2487 . 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 1369    e. wcel 1756   E.wrex 2735   {crab 2738   _Vcvv 2991    u. cun 3345   ~Pcpw 3879   class class class wbr 4311   ` cfv 5437  (class class class)co 6110    e. cmpt2 6112   lecple 14264   joincjn 15133   Atomscatm 32931   +Pcpadd 33462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597  df-padd 33463
This theorem is referenced by:  paddval  33465
  Copyright terms: Public domain W3C validator