Theorem nvscl

Description: Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvscl.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	nvscl.4	$$⊢S={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
Assertion	nvscl	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in ℂ\wedge B\in X\right)\to ASB\in X$$

Proof

Step	Hyp	Ref	Expression
1		nvscl.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		nvscl.4	$$⊢S={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
3		eqid	$$⊢1^{st}\left(U\right)=1^{st}\left(U\right)$$
4	3	nvvc	$$⊢U\in \mathrm{NrmCVec}\to 1^{st}\left(U\right)\in {CVec}_{\mathrm{OLD}}$$
5		eqid	$$⊢{+}_v\left(U\right)={+}_v\left(U\right)$$
6	5	vafval	$$⊢{+}_v\left(U\right)=1^{st}\left(1^{st}\left(U\right)\right)$$
7	2	smfval	$$⊢S=2^{nd}\left(1^{st}\left(U\right)\right)$$
8	1 5	bafval	$$⊢X=\mathrm{ran}{+}_v\left(U\right)$$
9	6 7 8	vccl	$$⊢\left(1^{st}\left(U\right)\in {CVec}_{\mathrm{OLD}}\wedge A\in ℂ\wedge B\in X\right)\to ASB\in X$$
10	4 9	syl3an1	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in ℂ\wedge B\in X\right)\to ASB\in X$$