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Theorem pjpm 19263
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 2423 . . . . 5  |-  ( ocv `  W )  =  ( ocv `  W )
4 eqid 2423 . . . . 5  |-  ( proj1 `  W )  =  ( proj1 `  W )
5 pjpm.k . . . . 5  |-  K  =  ( proj `  W
)
61, 2, 3, 4, 5pjfval 19261 . . . 4  |-  K  =  ( ( x  e.  L  |->  ( x (
proj1 `  W
) ( ( ocv `  W ) `  x
) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
7 inss1 3683 . . . 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 3495 . . 3  |-  K  C_  ( x  e.  L  |->  ( x ( proj1 `  W )
( ( ocv `  W
) `  x )
) )
9 funmpt 5635 . . 3  |-  Fun  (
x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )
10 funss 5617 . . 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 2423 . . . . . 6  |-  ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )  =  ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )
13 ovex 6331 . . . . . . 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 6058 . . . . 5  |-  ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) ) : L --> _V
16 fssxp 5756 . . . . 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 3688 . . . . 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 3495 . . 3  |-  K  C_  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )
20 inxp 4984 . . . 4  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )
21 inv1 3790 . . . . 5  |-  ( L  i^i  _V )  =  L
22 incom 3656 . . . . . 6  |-  ( _V 
i^i  ( V  ^m  V ) )  =  ( ( V  ^m  V )  i^i  _V )
23 inv1 3790 . . . . . 6  |-  ( ( V  ^m  V )  i^i  _V )  =  ( V  ^m  V
)
2422, 23eqtri 2452 . . . . 5  |-  ( _V 
i^i  ( V  ^m  V ) )  =  ( V  ^m  V
)
2521, 24xpeq12i 4873 . . . 4  |-  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )  =  ( L  X.  ( V  ^m  V ) )
2620, 25eqtri 2452 . . 3  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( L  X.  ( V  ^m  V ) )
2719, 26sseqtri 3497 . 2  |-  K  C_  ( L  X.  ( V  ^m  V ) )
28 ovex 6331 . . 3  |-  ( V  ^m  V )  e. 
_V
29 fvex 5889 . . . 4  |-  ( LSubSp `  W )  e.  _V
302, 29eqeltri 2507 . . 3  |-  L  e. 
_V
3128, 30elpm 7508 . 2  |-  ( K  e.  ( ( V  ^m  V )  ^pm  L )  <->  ( Fun  K  /\  K  C_  ( L  X.  ( V  ^m  V ) ) ) )
3211, 27, 31mpbir2an 929 1  |-  K  e.  ( ( V  ^m  V )  ^pm  L
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1438    e. wcel 1869   _Vcvv 3082    i^i cin 3436    C_ wss 3437    |-> cmpt 4480    X. cxp 4849   Fun wfun 5593   -->wf 5595   ` cfv 5599  (class class class)co 6303    ^m cmap 7478    ^pm cpm 7479   Basecbs 15114   proj1cpj1 17280   LSubSpclss 18148   ocvcocv 19215   projcpj 19255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-pm 7481  df-pj 19258
This theorem is referenced by: (None)
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