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Theorem pjpm 16624
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 2296 . . . . 5  |-  ( ocv `  W )  =  ( ocv `  W )
4 eqid 2296 . . . . 5  |-  ( proj
1 `  W )  =  ( proj 1 `  W )
5 pjpm.k . . . . 5  |-  K  =  ( proj `  W
)
61, 2, 3, 4, 5pjfval 16622 . . . 4  |-  K  =  ( ( x  e.  L  |->  ( x (
proj 1 `  W ) ( ( ocv `  W
) `  x )
) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
7 inss1 3402 . . . 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 3221 . . 3  |-  K  C_  ( x  e.  L  |->  ( x ( proj
1 `  W )
( ( ocv `  W
) `  x )
) )
9 funmpt 5306 . . 3  |-  Fun  (
x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) )
10 funss 5289 . . 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 2296 . . . . . 6  |-  ( x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) )  =  ( x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) )
13 ovex 5899 . . . . . . 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 5699 . . . . 5  |-  ( x  e.  L  |->  ( x ( proj 1 `  W ) ( ( ocv `  W ) `
 x ) ) ) : L --> _V
16 fssxp 5416 . . . . 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 3407 . . . . 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 3221 . . 3  |-  K  C_  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )
20 inxp 4834 . . . 4  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )
21 inv1 3494 . . . . 5  |-  ( L  i^i  _V )  =  L
22 incom 3374 . . . . . 6  |-  ( _V 
i^i  ( V  ^m  V ) )  =  ( ( V  ^m  V )  i^i  _V )
23 inv1 3494 . . . . . 6  |-  ( ( V  ^m  V )  i^i  _V )  =  ( V  ^m  V
)
2422, 23eqtri 2316 . . . . 5  |-  ( _V 
i^i  ( V  ^m  V ) )  =  ( V  ^m  V
)
2521, 24xpeq12i 4727 . . . 4  |-  ( ( L  i^i  _V )  X.  ( _V  i^i  ( V  ^m  V ) ) )  =  ( L  X.  ( V  ^m  V ) )
2620, 25eqtri 2316 . . 3  |-  ( ( L  X.  _V )  i^i  ( _V  X.  ( V  ^m  V ) ) )  =  ( L  X.  ( V  ^m  V ) )
2719, 26sseqtri 3223 . 2  |-  K  C_  ( L  X.  ( V  ^m  V ) )
28 ovex 5899 . . 3  |-  ( V  ^m  V )  e. 
_V
29 fvex 5555 . . . 4  |-  ( LSubSp `  W )  e.  _V
302, 29eqeltri 2366 . . 3  |-  L  e. 
_V
3128, 30elpm 6814 . 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 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165    e. cmpt 4093    X. cxp 4703   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788    ^pm cpm 6789   Basecbs 13164   proj 1cpj1 14962   LSubSpclss 15705   ocvcocv 16576   projcpj 16616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-pm 6791  df-pj 16619
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