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Theorem lnopfi 29400
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 29290 . 2 (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
31, 2ax-mp 5 1 𝑇: ℋ⟶ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  wf 6131  chba 28348  LinOpclo 28376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-hilex 28428
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-map 8142  df-lnop 29272
This theorem is referenced by:  lnopaddi  29402  lnopsubi  29405  hoddii  29420  nmlnop0iALT  29426  nmlnopgt0i  29428  lnopmi  29431  lnophsi  29432  lnophdi  29433  lnopcoi  29434  lnopco0i  29435  lnopeq0lem1  29436  lnopeq0i  29438  lnopeqi  29439  lnopunilem1  29441  lnopunilem2  29442  lnophmlem2  29448  lnophmi  29449  nmbdoplbi  29455  nmcopexi  29458  nmcoplbi  29459  lnopconi  29465  imaelshi  29489  rnelshi  29490  cnlnadjlem2  29499  cnlnadjlem6  29503  cnlnadjlem7  29504  cnlnadjeui  29508  nmopcoi  29526  bdopcoi  29529  hmopidmchi  29582  hmopidmpji  29583
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