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

Theorem pjpm 16927
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 2435 . . . . 5  |-  ( ocv `  W )  =  ( ocv `  W )
4 eqid 2435 . . . . 5  |-  ( proj
1 `  W )  =  ( proj 1 `  W )
5 pjpm.k . . . . 5  |-  K  =  ( proj `  W
)
61, 2, 3, 4, 5pjfval 16925 . . . 4  |-  K  =  ( ( x  e.  L  |->  ( x (
proj 1 `  W ) ( ( ocv `  W
) `  x )
) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
7 inss1 3553 . . . 4  |-  ( ( x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) ) 
C_  ( x  e.  L  |->  ( x (
proj 1 `  W ) ( ( ocv `  W
) `  x )
) )
86, 7eqsstri 3370 . . 3  |-  K  C_  ( x  e.  L  |->  ( x ( proj
1 `  W )
( ( ocv `  W
) `  x )
) )
9 funmpt 5481 . . 3  |-  Fun  (
x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) )
10 funss 5464 . . 3  |-  ( K 
C_  ( x  e.  L  |->  ( x (
proj 1 `  W ) ( ( ocv `  W
) `  x )
) )  ->  ( Fun  ( x  e.  L  |->  ( x ( proj
1 `  W )
( ( ocv `  W
) `  x )
) )  ->  Fun  K ) )
118, 9, 10mp2 9 . 2  |-  Fun  K
12 eqid 2435 . . . . . 6  |-  ( x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) )  =  ( x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) )
13 ovex 6098 . . . . . . 7  |-  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) )  e.  _V
1413a1i 11 . . . . . 6  |-  ( x  e.  L  ->  (
x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) )  e.  _V )
1512, 14fmpti 5884 . . . . 5  |-  ( x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) ) : L --> _V
16 fssxp 5594 . . . . 5  |-  ( ( x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) ) : L --> _V  ->  ( x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) )  C_  ( L  X.  _V ) )
17 ssrin 3558 . . . . 5  |-  ( ( x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) )  C_  ( L  X.  _V )  ->  (
( x  e.  L  |->  ( x ( proj
1 `  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 ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) )  i^i  ( _V 
X.  ( V  ^m  V ) ) ) 
C_  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )
196, 18eqsstri 3370 . . 3  |-  K  C_  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )
20 inxp 4999 . . . 4  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )
21 inv1 3646 . . . . 5  |-  ( L  i^i  _V )  =  L
22 incom 3525 . . . . . 6  |-  ( _V 
i^i  ( V  ^m  V ) )  =  ( ( V  ^m  V )  i^i  _V )
23 inv1 3646 . . . . . 6  |-  ( ( V  ^m  V )  i^i  _V )  =  ( V  ^m  V
)
2422, 23eqtri 2455 . . . . 5  |-  ( _V 
i^i  ( V  ^m  V ) )  =  ( V  ^m  V
)
2521, 24xpeq12i 4892 . . . 4  |-  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )  =  ( L  X.  ( V  ^m  V ) )
2620, 25eqtri 2455 . . 3  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( L  X.  ( V  ^m  V ) )
2719, 26sseqtri 3372 . 2  |-  K  C_  ( L  X.  ( V  ^m  V ) )
28 ovex 6098 . . 3  |-  ( V  ^m  V )  e. 
_V
29 fvex 5734 . . . 4  |-  ( LSubSp `  W )  e.  _V
302, 29eqeltri 2505 . . 3  |-  L  e. 
_V
3128, 30elpm 7036 . 2  |-  ( K  e.  ( ( V  ^m  V )  ^pm  L )  <->  ( Fun  K  /\  K  C_  ( L  X.  ( V  ^m  V ) ) ) )
3211, 27, 31mpbir2an 887 1  |-  K  e.  ( ( V  ^m  V )  ^pm  L
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311    C_ wss 3312    e. cmpt 4258    X. cxp 4868   Fun wfun 5440   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^m cmap 7010    ^pm cpm 7011   Basecbs 13461   proj 1cpj1 15261   LSubSpclss 16000   ocvcocv 16879   projcpj 16919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-pm 7013  df-pj 16922
  Copyright terms: Public domain W3C validator