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Theorem pjpm 18155
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 2443 . . . . 5  |-  ( ocv `  W )  =  ( ocv `  W )
4 eqid 2443 . . . . 5  |-  ( proj1 `  W )  =  ( proj1 `  W )
5 pjpm.k . . . . 5  |-  K  =  ( proj `  W
)
61, 2, 3, 4, 5pjfval 18153 . . . 4  |-  K  =  ( ( x  e.  L  |->  ( x (
proj1 `  W
) ( ( ocv `  W ) `  x
) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
7 inss1 3591 . . . 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 3407 . . 3  |-  K  C_  ( x  e.  L  |->  ( x ( proj1 `  W )
( ( ocv `  W
) `  x )
) )
9 funmpt 5475 . . 3  |-  Fun  (
x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )
10 funss 5457 . . 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 2443 . . . . . 6  |-  ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )  =  ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) )
13 ovex 6137 . . . . . . 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 5887 . . . . 5  |-  ( x  e.  L  |->  ( x ( proj1 `  W ) ( ( ocv `  W ) `
 x ) ) ) : L --> _V
16 fssxp 5591 . . . . 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 3596 . . . . 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 3407 . . 3  |-  K  C_  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )
20 inxp 4993 . . . 4  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )
21 inv1 3685 . . . . 5  |-  ( L  i^i  _V )  =  L
22 incom 3564 . . . . . 6  |-  ( _V 
i^i  ( V  ^m  V ) )  =  ( ( V  ^m  V )  i^i  _V )
23 inv1 3685 . . . . . 6  |-  ( ( V  ^m  V )  i^i  _V )  =  ( V  ^m  V
)
2422, 23eqtri 2463 . . . . 5  |-  ( _V 
i^i  ( V  ^m  V ) )  =  ( V  ^m  V
)
2521, 24xpeq12i 4883 . . . 4  |-  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )  =  ( L  X.  ( V  ^m  V ) )
2620, 25eqtri 2463 . . 3  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( L  X.  ( V  ^m  V ) )
2719, 26sseqtri 3409 . 2  |-  K  C_  ( L  X.  ( V  ^m  V ) )
28 ovex 6137 . . 3  |-  ( V  ^m  V )  e. 
_V
29 fvex 5722 . . . 4  |-  ( LSubSp `  W )  e.  _V
302, 29eqeltri 2513 . . 3  |-  L  e. 
_V
3128, 30elpm 7264 . 2  |-  ( K  e.  ( ( V  ^m  V )  ^pm  L )  <->  ( Fun  K  /\  K  C_  ( L  X.  ( V  ^m  V ) ) ) )
3211, 27, 31mpbir2an 911 1  |-  K  e.  ( ( V  ^m  V )  ^pm  L
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   _Vcvv 2993    i^i cin 3348    C_ wss 3349    e. cmpt 4371    X. cxp 4859   Fun wfun 5433   -->wf 5435   ` cfv 5439  (class class class)co 6112    ^m cmap 7235    ^pm cpm 7236   Basecbs 14195   proj1cpj1 16155   LSubSpclss 17035   ocvcocv 18107   projcpj 18147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-pm 7238  df-pj 18150
This theorem is referenced by: (None)
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