Metamath Proof Explorer


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 = ⟨ ⟨ + , · ⟩ , norm
axhil.2 𝑈 ∈ CHilOLD
Assertion axhvmul0-zf ( 𝐴 ∈ ℋ → ( 0 · 𝐴 ) = 0 )

Proof

Step Hyp Ref Expression
1 axhil.1 𝑈 = ⟨ ⟨ + , · ⟩ , norm
2 axhil.2 𝑈 ∈ CHilOLD
3 df-hba ℋ = ( BaseSet ‘ ⟨ ⟨ + , · ⟩ , norm ⟩ )
4 1 fveq2i ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ ⟨ ⟨ + , · ⟩ , norm ⟩ )
5 3 4 eqtr4i ℋ = ( BaseSet ‘ 𝑈 )
6 2 hlnvi 𝑈 ∈ NrmCVec
7 1 6 h2hsm · = ( ·𝑠OLD𝑈 )
8 df-h0v 0 = ( 0vec ‘ ⟨ ⟨ + , · ⟩ , norm ⟩ )
9 1 fveq2i ( 0vec𝑈 ) = ( 0vec ‘ ⟨ ⟨ + , · ⟩ , norm ⟩ )
10 8 9 eqtr4i 0 = ( 0vec𝑈 )
11 5 7 10 hlmul0 ( ( 𝑈 ∈ CHilOLD𝐴 ∈ ℋ ) → ( 0 · 𝐴 ) = 0 )
12 2 11 mpan ( 𝐴 ∈ ℋ → ( 0 · 𝐴 ) = 0 )