MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-pj Structured version   Unicode version

Definition df-pj 19258
Description: Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 17282, 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 ( proj1 `  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 19255 . 2  class  proj
2 vh . . 3  setvar  h
3 cvv 3082 . . 3  class  _V
4 vx . . . . 5  setvar  x
52cv 1437 . . . . . 6  class  h
6 clss 18148 . . . . . 6  class  LSubSp
75, 6cfv 5599 . . . . 5  class  ( LSubSp `  h )
84cv 1437 . . . . . 6  class  x
9 cocv 19215 . . . . . . . 8  class  ocv
105, 9cfv 5599 . . . . . . 7  class  ( ocv `  h )
118, 10cfv 5599 . . . . . 6  class  ( ( ocv `  h ) `
 x )
12 cpj1 17280 . . . . . . 7  class  proj1
135, 12cfv 5599 . . . . . 6  class  ( proj1 `  h )
148, 11, 13co 6303 . . . . 5  class  ( x ( proj1 `  h ) ( ( ocv `  h ) `
 x ) )
154, 7, 14cmpt 4480 . . . 4  class  ( x  e.  ( LSubSp `  h
)  |->  ( x (
proj1 `  h
) ( ( ocv `  h ) `  x
) ) )
16 cbs 15114 . . . . . . 7  class  Base
175, 16cfv 5599 . . . . . 6  class  ( Base `  h )
18 cmap 7478 . . . . . 6  class  ^m
1917, 17, 18co 6303 . . . . 5  class  ( (
Base `  h )  ^m  ( Base `  h
) )
203, 19cxp 4849 . . . 4  class  ( _V 
X.  ( ( Base `  h )  ^m  ( Base `  h ) ) )
2115, 20cin 3436 . . 3  class  ( ( x  e.  ( LSubSp `  h )  |->  ( x ( proj1 `  h ) ( ( ocv `  h ) `
 x ) ) )  i^i  ( _V 
X.  ( ( Base `  h )  ^m  ( Base `  h ) ) ) )
222, 3, 21cmpt 4480 . 2  class  ( h  e.  _V  |->  ( ( x  e.  ( LSubSp `  h )  |->  ( x ( proj1 `  h ) ( ( ocv `  h ) `
 x ) ) )  i^i  ( _V 
X.  ( ( Base `  h )  ^m  ( Base `  h ) ) ) ) )
231, 22wceq 1438 1  wff  proj  =  ( h  e.  _V  |->  ( ( x  e.  ( LSubSp `  h )  |->  ( x ( proj1 `  h )
( ( ocv `  h
) `  x )
) )  i^i  ( _V  X.  ( ( Base `  h )  ^m  ( Base `  h ) ) ) ) )
Colors of variables: wff setvar class
This definition is referenced by:  pjfval  19261
  Copyright terms: Public domain W3C validator