Theorem cnlnadjlem1 28926

Description: Lemma for cnlnadji 28935 (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. ContOp
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

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 |- ( g = A -> ( T ` g ) = ( T ` A ) )
2	1	oveq1d 6665	. 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 6678	. 2 |- ( ( T ` A ) .ih y ) e. _V
5	2, 3, 4	fvmpt 6282	1 |- ( A e. ~H -> ( G ` A ) = ( ( T ` A ) .ih y ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   ~Hchil 27776   .ih csp 27779   ContOpccop 27803   LinOpclo 27804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  cnlnadjlem2  28927  cnlnadjlem3  28928  cnlnadjlem5  28930