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Theorem bdopf 29553
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 29552 . 2 (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)
2 lnopf 29550 . 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 2107  wf 6348  chba 28610  LinOpclo 28638  BndLinOpcbo 28639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-hilex 28690
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8398  df-lnop 29532  df-bdop 29533
This theorem is referenced by:  nmopre  29561  nmophmi  29722  adjbdln  29774  nmopadjlem  29780  nmoptrii  29785  nmopcoi  29786  bdophsi  29787  bdophdi  29788  nmoptri2i  29790  adjcoi  29791  nmopcoadji  29792  unierri  29795
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