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

Theorem pjpm 16608
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 2283 . . . . 5  |-  ( ocv `  W )  =  ( ocv `  W )
4 eqid 2283 . . . . 5  |-  ( proj
1 `  W )  =  ( proj 1 `  W )
5 pjpm.k . . . . 5  |-  K  =  ( proj `  W
)
61, 2, 3, 4, 5pjfval 16606 . . . 4  |-  K  =  ( ( x  e.  L  |->  ( x (
proj 1 `  W ) ( ( ocv `  W
) `  x )
) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
7 inss1 3389 . . . 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 3208 . . 3  |-  K  C_  ( x  e.  L  |->  ( x ( proj
1 `  W )
( ( ocv `  W
) `  x )
) )
9 funmpt 5290 . . 3  |-  Fun  (
x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) )
10 funss 5273 . . 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 17 . 2  |-  Fun  K
12 eqid 2283 . . . . . 6  |-  ( x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) )  =  ( x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) )
13 ovex 5883 . . . . . . 7  |-  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) )  e.  _V
1413a1i 10 . . . . . 6  |-  ( x  e.  L  ->  (
x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) )  e.  _V )
1512, 14fmpti 5683 . . . . 5  |-  ( x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) ) : L --> _V
16 fssxp 5400 . . . . 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 3394 . . . . 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 9 . . . 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 3208 . . 3  |-  K  C_  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )
20 inxp 4818 . . . 4  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )
21 inv1 3481 . . . . 5  |-  ( L  i^i  _V )  =  L
22 incom 3361 . . . . . 6  |-  ( _V 
i^i  ( V  ^m  V ) )  =  ( ( V  ^m  V )  i^i  _V )
23 inv1 3481 . . . . . 6  |-  ( ( V  ^m  V )  i^i  _V )  =  ( V  ^m  V
)
2422, 23eqtri 2303 . . . . 5  |-  ( _V 
i^i  ( V  ^m  V ) )  =  ( V  ^m  V
)
2521, 24xpeq12i 4711 . . . 4  |-  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )  =  ( L  X.  ( V  ^m  V ) )
2620, 25eqtri 2303 . . 3  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( L  X.  ( V  ^m  V ) )
2719, 26sseqtri 3210 . 2  |-  K  C_  ( L  X.  ( V  ^m  V ) )
28 ovex 5883 . . 3  |-  ( V  ^m  V )  e. 
_V
29 fvex 5539 . . . 4  |-  ( LSubSp `  W )  e.  _V
302, 29eqeltri 2353 . . 3  |-  L  e. 
_V
3128, 30elpm 6798 . 2  |-  ( K  e.  ( ( V  ^m  V )  ^pm  L )  <->  ( Fun  K  /\  K  C_  ( L  X.  ( V  ^m  V ) ) ) )
3211, 27, 31mpbir2an 886 1  |-  K  e.  ( ( V  ^m  V )  ^pm  L
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152    e. cmpt 4077    X. cxp 4687   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772    ^pm cpm 6773   Basecbs 13148   proj 1cpj1 14946   LSubSpclss 15689   ocvcocv 16560   projcpj 16600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-pm 6775  df-pj 16603
  Copyright terms: Public domain W3C validator