Theorem axhilex 8851

Description: Derive axiom ax-hilex 8869 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 8851 through axhcompl 8868, using ZFC set theory only. These show that if we are given a known, fixed Hilbert space U = ⟨⟨ +_h , ·_h ⟩, norm_h⟩ 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 8837 above. See also the comment in ax-hilex 8869.

Hypotheses

Ref	Expression
axhil.1	⊢ U = ⟨⟨ +h , ·h ⟩, normh⟩
axhil.2	⊢ U ∈ CHil

Assertion

Ref	Expression
axhilex	⊢ ℋ ∈ V

Proof of Theorem axhilex

Step	Hyp	Ref	Expression
1		df-hba 8838	. 2 ⊢ ℋ = (Base ‘⟨⟨ +h , ·h ⟩, normh⟩)
2	1	hlex 8600	1 ⊢ ℋ ∈ V

Syntax hints:   = wceq 956   ∈ wcel 958  Vcvv 1811  ⟨cop 2411  CHilchl 8589   ℋ chil 8788   +_h cva 8789   ·_h csm 8790  norm_hcno 8794

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-uni 2504  df-fv 3198  df-hba 8838

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