Theorem hldir

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

	Ref	Expression
Hypotheses	hldi.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	hldi.2	$$⊢G={+}_v\left(U\right)$$
	hldi.4	$$⊢S={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
Assertion	hldir	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge \left(A\in ℂ\wedge B\in ℂ\wedge C\in X\right)\right)\to \left(A+B\right)SC=\left(ASC\right)G\left(BSC\right)$$

Proof

Step	Hyp	Ref	Expression
1		hldi.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		hldi.2	$$⊢G={+}_v\left(U\right)$$
3		hldi.4	$$⊢S={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
4		hlnv	$$⊢U\in {CHil}_{\mathrm{OLD}}\to U\in \mathrm{NrmCVec}$$
5	1 2 3	nvdir	$$⊢\left(U\in \mathrm{NrmCVec}\wedge \left(A\in ℂ\wedge B\in ℂ\wedge C\in X\right)\right)\to \left(A+B\right)SC=\left(ASC\right)G\left(BSC\right)$$
6	4 5	sylan	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge \left(A\in ℂ\wedge B\in ℂ\wedge C\in X\right)\right)\to \left(A+B\right)SC=\left(ASC\right)G\left(BSC\right)$$