Theorem axhilex 8846

Description: Derive axiom ax-hilex 8864 from Hilbert space under ZF set theory.

Before introducing the 18 axioms for Hilbert space, we first prove them as the conclusions of theorems axhilex 8846 through axhcompl 8863, using ZFC set theory only. These show that if we are given a known, fixed Hilbert space U = <.<. +h , .h >., normh>. that satisfies their hypotheses, then we can derive the Hilbert space axioms as theorems of ZFC set theory. In practice, in order to use these theorems to convert the Hilbert Space explorer to a ZFC-only subtheory, we would also have to provide definitions for the 3 (otherwise primitive) class constants +h, .h, and .ih before df-hnorm 8832 above. See also the comment in ax-hilex 8864.

Hypotheses

Ref	Expression
axhil.1	|- U = <.<. +h , .h >., normh>.
axhil.2	|- U e. CHil

Assertion

Ref	Expression
axhilex	|- H~ e. V

Proof of Theorem axhilex

Step	Hyp	Ref	Expression
1		df-hba 8833	. 2 |- H~ = (Base` <.<. +h , .h >., normh>.)
2	1	hlex 8596	1 |- H~ e. V

Syntax hints:   = wceq 958   e. wcel 960  Vcvv 1814  <.cop 2415  CHilchl 8585  H~chil 8783   +h cva 8784   .h csm 8785  normhcno 8789

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-uni 2508  df-fv 3204  df-hba 8833

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