Theorem hlmul0

Description: Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hlmul0.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	hlmul0.4	$$⊢S={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
	hlmul0.5	$$⊢Z=0_{\mathrm{vec}}\left(U\right)$$
Assertion	hlmul0	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge A\in X\right)\to 0SA=Z$$

Proof

Step	Hyp	Ref	Expression
1		hlmul0.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		hlmul0.4	$$⊢S={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
3		hlmul0.5	$$⊢Z=0_{\mathrm{vec}}\left(U\right)$$
4		hlnv	$$⊢U\in {CHil}_{\mathrm{OLD}}\to U\in \mathrm{NrmCVec}$$
5	1 2 3	nv0	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\right)\to 0SA=Z$$
6	4 5	sylan	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge A\in X\right)\to 0SA=Z$$