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Theorem bdopf 29566
Description: A bounded linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
bdopf (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)

Proof of Theorem bdopf
StepHypRef Expression
1 bdopln 29565 . 2 (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)
2 lnopf 29563 . 2 (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
31, 2syl 17 1 (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wf 6344  chba 28623  LinOpclo 28651  BndLinOpcbo 28652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-hilex 28703
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-lnop 29545  df-bdop 29546
This theorem is referenced by:  nmopre  29574  nmophmi  29735  adjbdln  29787  nmopadjlem  29793  nmoptrii  29798  nmopcoi  29799  bdophsi  29800  bdophdi  29801  nmoptri2i  29803  adjcoi  29804  nmopcoadji  29805  unierri  29808
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