Metamath Proof Explorer


Theorem axhis2-zf

Description: Derive Axiom ax-his2 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
axhfi.1 ·ih = ( ·𝑖OLD𝑈 )
Assertion axhis2-zf ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 + 𝐵 ) ·ih 𝐶 ) = ( ( 𝐴 ·ih 𝐶 ) + ( 𝐵 ·ih 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 axhil.1 𝑈 = ⟨ ⟨ + , · ⟩ , norm
2 axhil.2 𝑈 ∈ CHilOLD
3 axhfi.1 ·ih = ( ·𝑖OLD𝑈 )
4 df-hba ℋ = ( BaseSet ‘ ⟨ ⟨ + , · ⟩ , norm ⟩ )
5 1 fveq2i ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ ⟨ ⟨ + , · ⟩ , norm ⟩ )
6 4 5 eqtr4i ℋ = ( BaseSet ‘ 𝑈 )
7 2 hlnvi 𝑈 ∈ NrmCVec
8 1 7 h2hva + = ( +𝑣𝑈 )
9 6 8 3 hlipdir ( ( 𝑈 ∈ CHilOLD ∧ ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) ) → ( ( 𝐴 + 𝐵 ) ·ih 𝐶 ) = ( ( 𝐴 ·ih 𝐶 ) + ( 𝐵 ·ih 𝐶 ) ) )
10 2 9 mpan ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 + 𝐵 ) ·ih 𝐶 ) = ( ( 𝐴 ·ih 𝐶 ) + ( 𝐵 ·ih 𝐶 ) ) )