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Theorem cnlnadjlem1 31058
Description: Lemma for cnlnadji 31067 (Theorem 3.10 of [Beran] p. 104: every continuous linear operator has an adjoint). The value of the auxiliary functional ๐บ. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
cnlnadjlem.1 ๐‘‡ โˆˆ LinOp
cnlnadjlem.2 ๐‘‡ โˆˆ ContOp
cnlnadjlem.3 ๐บ = (๐‘” โˆˆ โ„‹ โ†ฆ ((๐‘‡โ€˜๐‘”) ยทih ๐‘ฆ))
Assertion
Ref Expression
cnlnadjlem1 (๐ด โˆˆ โ„‹ โ†’ (๐บโ€˜๐ด) = ((๐‘‡โ€˜๐ด) ยทih ๐‘ฆ))
Distinct variable groups:   ๐‘ฆ,๐‘”,๐ด   ๐‘‡,๐‘”,๐‘ฆ
Allowed substitution hints:   ๐บ(๐‘ฆ,๐‘”)

Proof of Theorem cnlnadjlem1
StepHypRef Expression
1 fveq2 6846 . . 3 (๐‘” = ๐ด โ†’ (๐‘‡โ€˜๐‘”) = (๐‘‡โ€˜๐ด))
21oveq1d 7376 . 2 (๐‘” = ๐ด โ†’ ((๐‘‡โ€˜๐‘”) ยทih ๐‘ฆ) = ((๐‘‡โ€˜๐ด) ยทih ๐‘ฆ))
3 cnlnadjlem.3 . 2 ๐บ = (๐‘” โˆˆ โ„‹ โ†ฆ ((๐‘‡โ€˜๐‘”) ยทih ๐‘ฆ))
4 ovex 7394 . 2 ((๐‘‡โ€˜๐ด) ยทih ๐‘ฆ) โˆˆ V
52, 3, 4fvmpt 6952 1 (๐ด โˆˆ โ„‹ โ†’ (๐บโ€˜๐ด) = ((๐‘‡โ€˜๐ด) ยทih ๐‘ฆ))
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   = wceq 1542   โˆˆ wcel 2107   โ†ฆ cmpt 5192  โ€˜cfv 6500  (class class class)co 7361   โ„‹chba 29910   ยทih csp 29913  ContOpccop 29937  LinOpclo 29938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364
This theorem is referenced by:  cnlnadjlem2  31059  cnlnadjlem3  31060  cnlnadjlem5  31062
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