Theorem hhims2

Description: Hilbert space distance metric. (Contributed by NM, 10-Apr-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hhnv.1	$$⊢U=⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩$$
	hhims2.2	$$⊢D=\mathrm{IndMet}\left(U\right)$$
Assertion	hhims2	$$⊢D={norm}_{ℎ}\circ {-}_{ℎ}$$

Proof

Step	Hyp	Ref	Expression
1		hhnv.1	$$⊢U=⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩$$
2		hhims2.2	$$⊢D=\mathrm{IndMet}\left(U\right)$$
3		eqid	$$⊢{norm}_{ℎ}\circ {-}_{ℎ}={norm}_{ℎ}\circ {-}_{ℎ}$$
4	1 3	hhims	$$⊢{norm}_{ℎ}\circ {-}_{ℎ}=\mathrm{IndMet}\left(U\right)$$
5	2 4	eqtr4i	$$⊢D={norm}_{ℎ}\circ {-}_{ℎ}$$