Theorem hlmulid

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

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

Proof

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