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Theorem bdopf 28105
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 28104 . 2 (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)
2 lnopf 28102 . 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 1977  wf 5800  chil 27160  LinOpclo 27188  BndLinOpcbo 27189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-hilex 27240
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-lnop 28084  df-bdop 28085
This theorem is referenced by:  nmopre  28113  nmophmi  28274  adjbdln  28326  nmopadjlem  28332  nmoptrii  28337  nmopcoi  28338  bdophsi  28339  bdophdi  28340  nmoptri2i  28342  adjcoi  28343  nmopcoadji  28344  unierri  28347
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