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Theorem cnlnadjlem1 22647
Description: Lemma for cnlnadji 22656 (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 5525 . . 3  |-  ( g  =  A  ->  ( T `  g )  =  ( T `  A ) )
21oveq1d 5873 . 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 5883 . 2  |-  ( ( T `  A ) 
.ih  y )  e. 
_V
52, 3, 4fvmpt 5602 1  |-  ( A  e.  ~H  ->  ( G `  A )  =  ( ( T `
 A )  .ih  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   ~Hchil 21499    .ih csp 21502   ConOpccop 21526   LinOpclo 21527
This theorem is referenced by:  cnlnadjlem2  22648  cnlnadjlem3  22649  cnlnadjlem5  22651
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-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-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861
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