Metamath Proof Explorer


Theorem axhfvmul-zf

Description: Derive Axiom ax-hfvmul 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 axhfvmul-zf · : ( ℂ × ℋ ) ⟶ ℋ

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 5 7 hlmulf ( 𝑈 ∈ CHilOLD· : ( ℂ × ℋ ) ⟶ ℋ )
9 2 8 ax-mp · : ( ℂ × ℋ ) ⟶ ℋ