Metamath Proof Explorer


Theorem axhvass-zf

Description: Derive Axiom ax-hvass 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 axhvass-zf ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 + 𝐵 ) + 𝐶 ) = ( 𝐴 + ( 𝐵 + 𝐶 ) ) )

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 h2hva + = ( +𝑣𝑈 )
8 5 7 hlass ( ( 𝑈 ∈ CHilOLD ∧ ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) ) → ( ( 𝐴 + 𝐵 ) + 𝐶 ) = ( 𝐴 + ( 𝐵 + 𝐶 ) ) )
9 2 8 mpan ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 + 𝐵 ) + 𝐶 ) = ( 𝐴 + ( 𝐵 + 𝐶 ) ) )