Theorem meteq0

Description: The value of a metric is zero iff its arguments are equal. Property M2 of Kreyszig p. 4. (Contributed by NM, 30-Aug-2006)

	Ref	Expression
Assertion	meteq0	$$⊢\left(D\in \mathrm{Met}\left(X\right)\wedge A\in X\wedge B\in X\right)\to \left(ADB=0\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		metxmet	$$⊢D\in \mathrm{Met}\left(X\right)\to D\in \mathrm{\infty Met}\left(X\right)$$
2		xmeteq0	$$⊢\left(D\in \mathrm{\infty Met}\left(X\right)\wedge A\in X\wedge B\in X\right)\to \left(ADB=0\leftrightarrow A=B\right)$$
3	1 2	syl3an1	$$⊢\left(D\in \mathrm{Met}\left(X\right)\wedge A\in X\wedge B\in X\right)\to \left(ADB=0\leftrightarrow A=B\right)$$