Theorem axhvmul0-zf

Description: Derive Axiom ax-hvmul0 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	axhvmul0-zf	⊢ ( 𝐴 ∈ ℋ → ( 0 ·ℎ 𝐴 ) = 0ℎ )

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	2	hlnvi	⊢ 𝑈 ∈ NrmCVec
7	1 6	h2hsm	⊢ ·ℎ = ( ·𝑠OLD ‘ 𝑈 )
8		df-h0v	⊢ 0ℎ = ( 0vec ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
9	1	fveq2i	⊢ ( 0vec ‘ 𝑈 ) = ( 0vec ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
10	8 9	eqtr4i	⊢ 0ℎ = ( 0vec ‘ 𝑈 )
11	5 7 10	hlmul0	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ ℋ ) → ( 0 ·ℎ 𝐴 ) = 0ℎ )
12	2 11	mpan	⊢ ( 𝐴 ∈ ℋ → ( 0 ·ℎ 𝐴 ) = 0ℎ )