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Definition df-padd 29264
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 29263 . 2  class  + P
2 vl . . 3  set  l
3 cvv 2789 . . 3  class  _V
4 vm . . . 4  set  m
5 vn . . . 4  set  n
62cv 1622 . . . . . 6  class  l
7 catm 28732 . . . . . 6  class  Atoms
86, 7cfv 5221 . . . . 5  class  ( Atoms `  l )
98cpw 3626 . . . 4  class  ~P ( Atoms `  l )
104cv 1622 . . . . . 6  class  m
115cv 1622 . . . . . 6  class  n
1210, 11cun 3151 . . . . 5  class  ( m  u.  n )
13 vp . . . . . . . . . 10  set  p
1413cv 1622 . . . . . . . . 9  class  p
15 vq . . . . . . . . . . 11  set  q
1615cv 1622 . . . . . . . . . 10  class  q
17 vr . . . . . . . . . . 11  set  r
1817cv 1622 . . . . . . . . . 10  class  r
19 cjn 14074 . . . . . . . . . . 11  class  join
206, 19cfv 5221 . . . . . . . . . 10  class  ( join `  l )
2116, 18, 20co 5820 . . . . . . . . 9  class  ( q ( join `  l
) r )
22 cple 13211 . . . . . . . . . 10  class  le
236, 22cfv 5221 . . . . . . . . 9  class  ( le
`  l )
2414, 21, 23wbr 4024 . . . . . . . 8  wff  p ( le `  l ) ( q ( join `  l ) r )
2524, 17, 11wrex 2545 . . . . . . 7  wff  E. r  e.  n  p ( le `  l ) ( q ( join `  l
) r )
2625, 15, 10wrex 2545 . . . . . 6  wff  E. q  e.  m  E. r  e.  n  p ( le `  l ) ( q ( join `  l
) r )
2726, 13, 8crab 2548 . . . . 5  class  { p  e.  ( Atoms `  l )  |  E. q  e.  m  E. r  e.  n  p ( le `  l ) ( q ( join `  l
) r ) }
2812, 27cun 3151 . . . 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 5822 . . 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 4078 . 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 1623 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  29265
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