Theorem hhsm

Description: The scalar product operation of Hilbert space. (Contributed by NM, 17-Nov-2007) (New usage is discouraged.)

	Ref	Expression
Hypothesis	hhnv.1	$$⊢U=⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩$$
Assertion	hhsm	$$⊢{\cdot }_{ℎ}={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$

Proof

Step	Hyp	Ref	Expression
1		hhnv.1	$$⊢U=⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩$$
2	1	hhnv	$$⊢U\in \mathrm{NrmCVec}$$
3	1 2	h2hsm	$$⊢{\cdot }_{ℎ}={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$