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Theorem cnlnadjlem1 22663
Description: Lemma for cnlnadji 22672 (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 5541 . . 3  |-  ( g  =  A  ->  ( T `  g )  =  ( T `  A ) )
21oveq1d 5889 . 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 5899 . 2  |-  ( ( T `  A ) 
.ih  y )  e. 
_V
52, 3, 4fvmpt 5618 1  |-  ( A  e.  ~H  ->  ( G `  A )  =  ( ( T `
 A )  .ih  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   ~Hchil 21515    .ih csp 21518   ConOpccop 21542   LinOpclo 21543
This theorem is referenced by:  cnlnadjlem2  22664  cnlnadjlem3  22665  cnlnadjlem5  22667
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-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-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877
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