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Theorem cnlnadjlem1 22572
Description: Lemma for cnlnadji 22581 (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 5423 . . 3  |-  ( g  =  A  ->  ( T `  g )  =  ( T `  A ) )
21oveq1d 5772 . 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 5782 . 2  |-  ( ( T `  A ) 
.ih  y )  e. 
_V
52, 3, 4fvmpt 5501 1  |-  ( A  e.  ~H  ->  ( G `  A )  =  ( ( T `
 A )  .ih  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621    e. cmpt 4017   ` cfv 4638  (class class class)co 5757   ~Hchil 21424    .ih csp 21427   ConOpccop 21451   LinOpclo 21452
This theorem is referenced by:  cnlnadjlem2  22573  cnlnadjlem3  22574  cnlnadjlem5  22576
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654  df-ov 5760
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