HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  cnlnadjlem1 Unicode version

Theorem cnlnadjlem1 23523
Description: Lemma for cnlnadji 23532 (Theorem 3.10 of [Beran] p. 104: every continuous linear operator has an adjoint). The value of the auxiliary functional  G. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
cnlnadjlem.1  |-  T  e. 
LinOp
cnlnadjlem.2  |-  T  e. 
ConOp
cnlnadjlem.3  |-  G  =  ( g  e.  ~H  |->  ( ( T `  g )  .ih  y
) )
Assertion
Ref Expression
cnlnadjlem1  |-  ( A  e.  ~H  ->  ( G `  A )  =  ( ( T `
 A )  .ih  y ) )
Distinct variable groups:    y, g, A    T, g, y
Allowed substitution hints:    G( y, g)

Proof of Theorem cnlnadjlem1
StepHypRef Expression
1 fveq2 5687 . . 3  |-  ( g  =  A  ->  ( T `  g )  =  ( T `  A ) )
21oveq1d 6055 . 2  |-  ( g  =  A  ->  (
( T `  g
)  .ih  y )  =  ( ( T `
 A )  .ih  y ) )
3 cnlnadjlem.3 . 2  |-  G  =  ( g  e.  ~H  |->  ( ( T `  g )  .ih  y
) )
4 ovex 6065 . 2  |-  ( ( T `  A ) 
.ih  y )  e. 
_V
52, 3, 4fvmpt 5765 1  |-  ( A  e.  ~H  ->  ( G `  A )  =  ( ( T `
 A )  .ih  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   ~Hchil 22375    .ih csp 22378   ConOpccop 22402   LinOpclo 22403
This theorem is referenced by:  cnlnadjlem2  23524  cnlnadjlem3  23525  cnlnadjlem5  23527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043
  Copyright terms: Public domain W3C validator