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

Theorem pjpm 18534
Description: The projection map is a partial function from subspaces of the pre-Hilbert space to total operators. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjpm.v  |-  V  =  ( Base `  W
)
pjpm.l  |-  L  =  ( LSubSp `  W )
pjpm.k  |-  K  =  ( proj `  W
)
Assertion
Ref Expression
pjpm  |-  K  e.  ( ( V  ^m  V )  ^pm  L
)

Proof of Theorem pjpm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pjpm.v . . . . 5  |-  V  =  ( Base `  W
)
2 pjpm.l . . . . 5  |-  L  =  ( LSubSp `  W )
3 eqid 2467 . . . . 5  |-  ( ocv `  W )  =  ( ocv `  W )
4 eqid 2467 . . . . 5  |-  ( proj1 `  W )  =  ( proj1 `  W )
5 pjpm.k . . . . 5  |-  K  =  ( proj `  W
)
61, 2, 3, 4, 5pjfval 18532 . . . 4  |-  K  =  ( ( x  e.  L  |->  ( x (
proj1 `  W
) ( ( ocv `  W ) `  x
) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
7 inss1 3718 . . . 4  |-  ( ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) ) 
C_  ( x  e.  L  |->  ( x (
proj1 `  W
) ( ( ocv `  W ) `  x
) ) )
86, 7eqsstri 3534 . . 3  |-  K  C_  ( x  e.  L  |->  ( x ( proj1 `  W )
( ( ocv `  W
) `  x )
) )
9 funmpt 5624 . . 3  |-  Fun  (
x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )
10 funss 5606 . . 3  |-  ( K 
C_  ( x  e.  L  |->  ( x (
proj1 `  W
) ( ( ocv `  W ) `  x
) ) )  -> 
( Fun  ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )  ->  Fun  K ) )
118, 9, 10mp2 9 . 2  |-  Fun  K
12 eqid 2467 . . . . . 6  |-  ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )  =  ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )
13 ovex 6309 . . . . . . 7  |-  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) )  e.  _V
1413a1i 11 . . . . . 6  |-  ( x  e.  L  ->  (
x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) )  e.  _V )
1512, 14fmpti 6044 . . . . 5  |-  ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) ) : L --> _V
16 fssxp 5743 . . . . 5  |-  ( ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) ) : L --> _V  ->  ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )  C_  ( L  X.  _V ) )
17 ssrin 3723 . . . . 5  |-  ( ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )  C_  ( L  X.  _V )  ->  (
( x  e.  L  |->  ( x ( proj1 `  W )
( ( ocv `  W
) `  x )
) )  i^i  ( _V  X.  ( V  ^m  V ) ) ) 
C_  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) ) )
1815, 16, 17mp2b 10 . . . 4  |-  ( ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) ) 
C_  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )
196, 18eqsstri 3534 . . 3  |-  K  C_  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )
20 inxp 5135 . . . 4  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )
21 inv1 3812 . . . . 5  |-  ( L  i^i  _V )  =  L
22 incom 3691 . . . . . 6  |-  ( _V 
i^i  ( V  ^m  V ) )  =  ( ( V  ^m  V )  i^i  _V )
23 inv1 3812 . . . . . 6  |-  ( ( V  ^m  V )  i^i  _V )  =  ( V  ^m  V
)
2422, 23eqtri 2496 . . . . 5  |-  ( _V 
i^i  ( V  ^m  V ) )  =  ( V  ^m  V
)
2521, 24xpeq12i 5021 . . . 4  |-  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )  =  ( L  X.  ( V  ^m  V ) )
2620, 25eqtri 2496 . . 3  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( L  X.  ( V  ^m  V ) )
2719, 26sseqtri 3536 . 2  |-  K  C_  ( L  X.  ( V  ^m  V ) )
28 ovex 6309 . . 3  |-  ( V  ^m  V )  e. 
_V
29 fvex 5876 . . . 4  |-  ( LSubSp `  W )  e.  _V
302, 29eqeltri 2551 . . 3  |-  L  e. 
_V
3128, 30elpm 7449 . 2  |-  ( K  e.  ( ( V  ^m  V )  ^pm  L )  <->  ( Fun  K  /\  K  C_  ( L  X.  ( V  ^m  V ) ) ) )
3211, 27, 31mpbir2an 918 1  |-  K  e.  ( ( V  ^m  V )  ^pm  L
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476    |-> cmpt 4505    X. cxp 4997   Fun wfun 5582   -->wf 5584   ` cfv 5588  (class class class)co 6284    ^m cmap 7420    ^pm cpm 7421   Basecbs 14490   proj1cpj1 16461   LSubSpclss 17378   ocvcocv 18486   projcpj 18526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-pm 7423  df-pj 18529
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator