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

Theorem lnfnfi 31789
Description: A linear Hilbert space functional is a functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
lnfnl.1 𝑇 ∈ LinFn
Assertion
Ref Expression
lnfnfi 𝑇: β„‹βŸΆβ„‚

Proof of Theorem lnfnfi
StepHypRef Expression
1 lnfnl.1 . 2 𝑇 ∈ LinFn
2 lnfnf 31632 . 2 (𝑇 ∈ LinFn β†’ 𝑇: β„‹βŸΆβ„‚)
31, 2ax-mp 5 1 𝑇: β„‹βŸΆβ„‚
Colors of variables: wff setvar class
Syntax hints:   ∈ wcel 2098  βŸΆwf 6530  β„‚cc 11105   β„‹chba 30667  LinFnclf 30702
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-hilex 30747
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-lnfn 31596
This theorem is referenced by:  lnfn0i  31790  lnfnaddi  31791  lnfnmuli  31792  lnfnsubi  31794  nmbdfnlbi  31797  nmcfnexi  31799  nmcfnlbi  31800  lnfnconi  31803  nlelshi  31808  nlelchi  31809  riesz3i  31810  riesz4i  31811
  Copyright terms: Public domain W3C validator