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

Theorem axhilex-zf 28768
 Description: Derive axiom ax-hilex 28786 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
axhil.1 𝑈 = ⟨⟨ + , · ⟩, norm
axhil.2 𝑈 ∈ CHilOLD
Assertion
Ref Expression
axhilex-zf ℋ ∈ V

Proof of Theorem axhilex-zf
StepHypRef Expression
1 df-hba 28756 . 2 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
21hlex 28685 1 ℋ ∈ V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2112  Vcvv 3444  ⟨cop 4534  CHilOLDchlo 28672   ℋchba 28706   +ℎ cva 28707   ·ℎ csm 28708  normℎcno 28710 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-uni 4804  df-iota 6287  df-fv 6336  df-hba 28756 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator