Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  hocofi Structured version   Visualization version   GIF version

Theorem hocofi 29528
 Description: Mapping of composition of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
hoeq.1 𝑆: ℋ⟶ ℋ
hoeq.2 𝑇: ℋ⟶ ℋ
Assertion
Ref Expression
hocofi (𝑆𝑇): ℋ⟶ ℋ

Proof of Theorem hocofi
StepHypRef Expression
1 hoeq.1 . 2 𝑆: ℋ⟶ ℋ
2 hoeq.2 . 2 𝑇: ℋ⟶ ℋ
3 fco 6507 . 2 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆𝑇): ℋ⟶ ℋ)
41, 2, 3mp2an 690 1 (𝑆𝑇): ℋ⟶ ℋ
 Colors of variables: wff setvar class Syntax hints:   ∘ ccom 5535  ⟶wf 6327   ℋchba 28681 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-fun 6333  df-fn 6334  df-f 6335 This theorem is referenced by:  hocofni  29529  hocadddiri  29541  hocsubdiri  29542  ho2coi  29543  ho0coi  29550  hoid1i  29551  hoid1ri  29552  hoddii  29751  lnopcoi  29765  bdopcoi  29860  adjcoi  29862  nmopcoadji  29863  unierri  29866  pjsdii  29917  pjddii  29918  pjsdi2i  29919  pjss1coi  29925  pjss2coi  29926  pjorthcoi  29931  pjinvari  29953  pjclem1  29957  pjclem4  29961  pjadj2coi  29966  pj3lem1  29968  pj3si  29969  pj3cor1i  29971
 Copyright terms: Public domain W3C validator