Theorem axhv0cl-zf

Description: Derive Axiom ax-hv0cl from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	axhil.1	⊢ 𝑈 = ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩
	axhil.2	⊢ 𝑈 ∈ CHilOLD
Assertion	axhv0cl-zf	⊢ 0ℎ ∈ ℋ

Proof

Step	Hyp	Ref	Expression
1		axhil.1	⊢ 𝑈 = ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩
2		axhil.2	⊢ 𝑈 ∈ CHilOLD
3		df-hba	⊢ ℋ = ( BaseSet ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
4	1	fveq2i	⊢ ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
5	3 4	eqtr4i	⊢ ℋ = ( BaseSet ‘ 𝑈 )
6		df-h0v	⊢ 0ℎ = ( 0vec ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
7	1	fveq2i	⊢ ( 0vec ‘ 𝑈 ) = ( 0vec ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
8	6 7	eqtr4i	⊢ 0ℎ = ( 0vec ‘ 𝑈 )
9	5 8	hl0cl	⊢ ( 𝑈 ∈ CHilOLD → 0ℎ ∈ ℋ )
10	2 9	ax-mp	⊢ 0ℎ ∈ ℋ