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

Theorem axhilex-zf 27010
 Description: Derive axiom ax-hilex 27028 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 26998 . 2 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
21hlex 26926 1 ℋ ∈ V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1474   ∈ wcel 1938  Vcvv 3077  ⟨cop 4034  CHilOLDchlo 26913   ℋchil 26948   +ℎ cva 26949   ·ℎ csm 26950  normℎcno 26952 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-nul 4616 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-sn 4029  df-pr 4031  df-uni 4271  df-iota 5653  df-fv 5697  df-hba 26998 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator