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Theorem lnopfi 28698
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 28588 . 2 (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
31, 2ax-mp 5 1 𝑇: ℋ⟶ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  wf 5848  chil 27646  LinOpclo 27674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-hilex 27726
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-lnop 28570
This theorem is referenced by:  lnopaddi  28700  lnopsubi  28703  hoddii  28718  nmlnop0iALT  28724  nmlnopgt0i  28726  lnopmi  28729  lnophsi  28730  lnophdi  28731  lnopcoi  28732  lnopco0i  28733  lnopeq0lem1  28734  lnopeq0i  28736  lnopeqi  28737  lnopunilem1  28739  lnopunilem2  28740  lnophmlem2  28746  lnophmi  28747  nmbdoplbi  28753  nmcopexi  28756  nmcoplbi  28757  lnopconi  28763  imaelshi  28787  rnelshi  28788  cnlnadjlem2  28797  cnlnadjlem6  28801  cnlnadjlem7  28802  cnlnadjeui  28806  nmopcoi  28824  bdopcoi  28827  hmopidmchi  28880  hmopidmpji  28881
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