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Definition df-padd 29115
Description: Define projective sum of two subspaces (or more generally two sets of atoms), which is the union of all lines generated by pairs of atoms from each subspace. Lemma 16.2 of [MaedaMaeda] p. 68. For convenience, our definition is generalized to apply to empty sets. (Contributed by NM, 29-Dec-2011.)
Assertion
Ref Expression
df-padd  |-  + P  =  ( l  e. 
_V  |->  ( m  e. 
~P ( Atoms `  l
) ,  n  e. 
~P ( Atoms `  l
)  |->  ( ( m  u.  n )  u. 
{ p  e.  (
Atoms `  l )  |  E. q  e.  m  E. r  e.  n  p ( le `  l ) ( q ( join `  l
) r ) } ) ) )
Distinct variable group:    m, l, n, p, q, r

Detailed syntax breakdown of Definition df-padd
StepHypRef Expression
1 cpadd 29114 . 2  class  + P
2 vl . . 3  set  l
3 cvv 2740 . . 3  class  _V
4 vm . . . 4  set  m
5 vn . . . 4  set  n
62cv 1618 . . . . . 6  class  l
7 catm 28583 . . . . . 6  class  Atoms
86, 7cfv 4638 . . . . 5  class  ( Atoms `  l )
98cpw 3566 . . . 4  class  ~P ( Atoms `  l )
104cv 1618 . . . . . 6  class  m
115cv 1618 . . . . . 6  class  n
1210, 11cun 3092 . . . . 5  class  ( m  u.  n )
13 vp . . . . . . . . . 10  set  p
1413cv 1618 . . . . . . . . 9  class  p
15 vq . . . . . . . . . . 11  set  q
1615cv 1618 . . . . . . . . . 10  class  q
17 vr . . . . . . . . . . 11  set  r
1817cv 1618 . . . . . . . . . 10  class  r
19 cjn 14005 . . . . . . . . . . 11  class  join
206, 19cfv 4638 . . . . . . . . . 10  class  ( join `  l )
2116, 18, 20co 5757 . . . . . . . . 9  class  ( q ( join `  l
) r )
22 cple 13142 . . . . . . . . . 10  class  le
236, 22cfv 4638 . . . . . . . . 9  class  ( le
`  l )
2414, 21, 23wbr 3963 . . . . . . . 8  wff  p ( le `  l ) ( q ( join `  l ) r )
2524, 17, 11wrex 2517 . . . . . . 7  wff  E. r  e.  n  p ( le `  l ) ( q ( join `  l
) r )
2625, 15, 10wrex 2517 . . . . . 6  wff  E. q  e.  m  E. r  e.  n  p ( le `  l ) ( q ( join `  l
) r )
2726, 13, 8crab 2519 . . . . 5  class  { p  e.  ( Atoms `  l )  |  E. q  e.  m  E. r  e.  n  p ( le `  l ) ( q ( join `  l
) r ) }
2812, 27cun 3092 . . . 4  class  ( ( m  u.  n )  u.  { p  e.  ( Atoms `  l )  |  E. q  e.  m  E. r  e.  n  p ( le `  l ) ( q ( join `  l
) r ) } )
294, 5, 9, 9, 28cmpt2 5759 . . 3  class  ( m  e.  ~P ( Atoms `  l ) ,  n  e.  ~P ( Atoms `  l
)  |->  ( ( m  u.  n )  u. 
{ p  e.  (
Atoms `  l )  |  E. q  e.  m  E. r  e.  n  p ( le `  l ) ( q ( join `  l
) r ) } ) )
302, 3, 29cmpt 4017 . 2  class  ( l  e.  _V  |->  ( m  e.  ~P ( Atoms `  l ) ,  n  e.  ~P ( Atoms `  l
)  |->  ( ( m  u.  n )  u. 
{ p  e.  (
Atoms `  l )  |  E. q  e.  m  E. r  e.  n  p ( le `  l ) ( q ( join `  l
) r ) } ) ) )
311, 30wceq 1619 1  wff  + P  =  ( l  e. 
_V  |->  ( m  e. 
~P ( Atoms `  l
) ,  n  e. 
~P ( Atoms `  l
)  |->  ( ( m  u.  n )  u. 
{ p  e.  (
Atoms `  l )  |  E. q  e.  m  E. r  e.  n  p ( le `  l ) ( q ( join `  l
) r ) } ) ) )
Colors of variables: wff set class
This definition is referenced by:  paddfval  29116
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