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Definition df-pj 16603
Description: Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 14948, but we restrict the domain of this function to only total projection functions. (Contributed by Mario Carneiro, 16-Oct-2015.)
Assertion
Ref Expression
df-pj  |-  proj  =  ( h  e.  _V  |->  ( ( x  e.  ( LSubSp `  h )  |->  ( x ( proj
1 `  h )
( ( ocv `  h
) `  x )
) )  i^i  ( _V  X.  ( ( Base `  h )  ^m  ( Base `  h ) ) ) ) )
Distinct variable group:    x, h

Detailed syntax breakdown of Definition df-pj
StepHypRef Expression
1 cpj 16600 . 2  class  proj
2 vh . . 3  set  h
3 cvv 2788 . . 3  class  _V
4 vx . . . . 5  set  x
52cv 1622 . . . . . 6  class  h
6 clss 15689 . . . . . 6  class  LSubSp
75, 6cfv 5255 . . . . 5  class  ( LSubSp `  h )
84cv 1622 . . . . . 6  class  x
9 cocv 16560 . . . . . . . 8  class  ocv
105, 9cfv 5255 . . . . . . 7  class  ( ocv `  h )
118, 10cfv 5255 . . . . . 6  class  ( ( ocv `  h ) `
 x )
12 cpj1 14946 . . . . . . 7  class  proj 1
135, 12cfv 5255 . . . . . 6  class  ( proj
1 `  h )
148, 11, 13co 5858 . . . . 5  class  ( x ( proj 1 `  h ) ( ( ocv `  h ) `
 x ) )
154, 7, 14cmpt 4077 . . . 4  class  ( x  e.  ( LSubSp `  h
)  |->  ( x (
proj 1 `  h ) ( ( ocv `  h
) `  x )
) )
16 cbs 13148 . . . . . . 7  class  Base
175, 16cfv 5255 . . . . . 6  class  ( Base `  h )
18 cmap 6772 . . . . . 6  class  ^m
1917, 17, 18co 5858 . . . . 5  class  ( (
Base `  h )  ^m  ( Base `  h
) )
203, 19cxp 4687 . . . 4  class  ( _V 
X.  ( ( Base `  h )  ^m  ( Base `  h ) ) )
2115, 20cin 3151 . . 3  class  ( ( x  e.  ( LSubSp `  h )  |->  ( x ( proj 1 `  h ) ( ( ocv `  h ) `
 x ) ) )  i^i  ( _V 
X.  ( ( Base `  h )  ^m  ( Base `  h ) ) ) )
222, 3, 21cmpt 4077 . 2  class  ( h  e.  _V  |->  ( ( x  e.  ( LSubSp `  h )  |->  ( x ( proj 1 `  h ) ( ( ocv `  h ) `
 x ) ) )  i^i  ( _V 
X.  ( ( Base `  h )  ^m  ( Base `  h ) ) ) ) )
231, 22wceq 1623 1  wff  proj  =  ( h  e.  _V  |->  ( ( x  e.  ( LSubSp `  h )  |->  ( x ( proj
1 `  h )
( ( ocv `  h
) `  x )
) )  i^i  ( _V  X.  ( ( Base `  h )  ^m  ( Base `  h ) ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  pjfval  16606
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