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Theorem pjpm 19039
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 2404 . . . . 5  |-  ( ocv `  W )  =  ( ocv `  W )
4 eqid 2404 . . . . 5  |-  ( proj1 `  W )  =  ( proj1 `  W )
5 pjpm.k . . . . 5  |-  K  =  ( proj `  W
)
61, 2, 3, 4, 5pjfval 19037 . . . 4  |-  K  =  ( ( x  e.  L  |->  ( x (
proj1 `  W
) ( ( ocv `  W ) `  x
) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
7 inss1 3661 . . . 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 3474 . . 3  |-  K  C_  ( x  e.  L  |->  ( x ( proj1 `  W )
( ( ocv `  W
) `  x )
) )
9 funmpt 5607 . . 3  |-  Fun  (
x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )
10 funss 5589 . . 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 2404 . . . . . 6  |-  ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )  =  ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )
13 ovex 6308 . . . . . . 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 6034 . . . . 5  |-  ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) ) : L --> _V
16 fssxp 5728 . . . . 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 3666 . . . . 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 3474 . . 3  |-  K  C_  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )
20 inxp 4958 . . . 4  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )
21 inv1 3768 . . . . 5  |-  ( L  i^i  _V )  =  L
22 incom 3634 . . . . . 6  |-  ( _V 
i^i  ( V  ^m  V ) )  =  ( ( V  ^m  V )  i^i  _V )
23 inv1 3768 . . . . . 6  |-  ( ( V  ^m  V )  i^i  _V )  =  ( V  ^m  V
)
2422, 23eqtri 2433 . . . . 5  |-  ( _V 
i^i  ( V  ^m  V ) )  =  ( V  ^m  V
)
2521, 24xpeq12i 4847 . . . 4  |-  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )  =  ( L  X.  ( V  ^m  V ) )
2620, 25eqtri 2433 . . 3  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( L  X.  ( V  ^m  V ) )
2719, 26sseqtri 3476 . 2  |-  K  C_  ( L  X.  ( V  ^m  V ) )
28 ovex 6308 . . 3  |-  ( V  ^m  V )  e. 
_V
29 fvex 5861 . . . 4  |-  ( LSubSp `  W )  e.  _V
302, 29eqeltri 2488 . . 3  |-  L  e. 
_V
3128, 30elpm 7489 . 2  |-  ( K  e.  ( ( V  ^m  V )  ^pm  L )  <->  ( Fun  K  /\  K  C_  ( L  X.  ( V  ^m  V ) ) ) )
3211, 27, 31mpbir2an 923 1  |-  K  e.  ( ( V  ^m  V )  ^pm  L
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1407    e. wcel 1844   _Vcvv 3061    i^i cin 3415    C_ wss 3416    |-> cmpt 4455    X. cxp 4823   Fun wfun 5565   -->wf 5567   ` cfv 5571  (class class class)co 6280    ^m cmap 7459    ^pm cpm 7460   Basecbs 14843   proj1cpj1 16981   LSubSpclss 17900   ocvcocv 18991   projcpj 19031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-pm 7462  df-pj 19034
This theorem is referenced by: (None)
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