Theorem axhis4-zf

Description: Derive Axiom ax-his4 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
	axhfi.1	⊢ ·ih = ( ·𝑖OLD ‘ 𝑈 )
Assertion	axhis4-zf	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → 0 < ( 𝐴 ·ih 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		axhil.1	⊢ 𝑈 = ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩
2		axhil.2	⊢ 𝑈 ∈ CHilOLD
3		axhfi.1	⊢ ·ih = ( ·𝑖OLD ‘ 𝑈 )
4		df-hba	⊢ ℋ = ( BaseSet ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
5	1	fveq2i	⊢ ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
6	4 5	eqtr4i	⊢ ℋ = ( BaseSet ‘ 𝑈 )
7		df-h0v	⊢ 0ℎ = ( 0vec ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
8	1	fveq2i	⊢ ( 0vec ‘ 𝑈 ) = ( 0vec ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
9	7 8	eqtr4i	⊢ 0ℎ = ( 0vec ‘ 𝑈 )
10	6 9 3	hlipgt0	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → 0 < ( 𝐴 ·ih 𝐴 ) )
11	2 10	mp3an1	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → 0 < ( 𝐴 ·ih 𝐴 ) )