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Theorem lnopfi 29759
Description: A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
lnopl.1 𝑇 ∈ LinOp
Assertion
Ref Expression
lnopfi 𝑇: ℋ⟶ ℋ

Proof of Theorem lnopfi
StepHypRef Expression
1 lnopl.1 . 2 𝑇 ∈ LinOp
2 lnopf 29649 . 2 (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
31, 2ax-mp 5 1 𝑇: ℋ⟶ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 2115  wf 6339  chba 28709  LinOpclo 28737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-hilex 28789
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404  df-lnop 29631
This theorem is referenced by:  lnopaddi  29761  lnopsubi  29764  hoddii  29779  nmlnop0iALT  29785  nmlnopgt0i  29787  lnopmi  29790  lnophsi  29791  lnophdi  29792  lnopcoi  29793  lnopco0i  29794  lnopeq0lem1  29795  lnopeq0i  29797  lnopeqi  29798  lnopunilem1  29800  lnopunilem2  29801  lnophmlem2  29807  lnophmi  29808  nmbdoplbi  29814  nmcopexi  29817  nmcoplbi  29818  lnopconi  29824  imaelshi  29848  rnelshi  29849  cnlnadjlem2  29858  cnlnadjlem6  29862  cnlnadjlem7  29863  cnlnadjeui  29867  nmopcoi  29885  bdopcoi  29888  hmopidmchi  29941  hmopidmpji  29942
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