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Definition df-padd 29235
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 29234 . 2  class  + P
2 vl . . 3  set  l
3 cvv 2763 . . 3  class  _V
4 vm . . . 4  set  m
5 vn . . . 4  set  n
62cv 1618 . . . . . 6  class  l
7 catm 28703 . . . . . 6  class  Atoms
86, 7cfv 4673 . . . . 5  class  ( Atoms `  l )
98cpw 3599 . . . 4  class  ~P ( Atoms `  l )
104cv 1618 . . . . . 6  class  m
115cv 1618 . . . . . 6  class  n
1210, 11cun 3125 . . . . 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 14041 . . . . . . . . . . 11  class  join
206, 19cfv 4673 . . . . . . . . . 10  class  ( join `  l )
2116, 18, 20co 5792 . . . . . . . . 9  class  ( q ( join `  l
) r )
22 cple 13178 . . . . . . . . . 10  class  le
236, 22cfv 4673 . . . . . . . . 9  class  ( le
`  l )
2414, 21, 23wbr 3997 . . . . . . . 8  wff  p ( le `  l ) ( q ( join `  l ) r )
2524, 17, 11wrex 2519 . . . . . . 7  wff  E. r  e.  n  p ( le `  l ) ( q ( join `  l
) r )
2625, 15, 10wrex 2519 . . . . . 6  wff  E. q  e.  m  E. r  e.  n  p ( le `  l ) ( q ( join `  l
) r )
2726, 13, 8crab 2522 . . . . 5  class  { p  e.  ( Atoms `  l )  |  E. q  e.  m  E. r  e.  n  p ( le `  l ) ( q ( join `  l
) r ) }
2812, 27cun 3125 . . . 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 5794 . . 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 4051 . 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  29236
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