Theorem nvsf

Description: Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvsf.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	nvsf.4	$$⊢S={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
Assertion	nvsf	$$⊢U\in \mathrm{NrmCVec}\to S:ℂ\times X⟶X$$

Proof

Step	Hyp	Ref	Expression
1		nvsf.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		nvsf.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	vcsm	$$⊢1^{st}\left(U\right)\in {CVec}_{\mathrm{OLD}}\to S:ℂ\times X⟶X$$
10	4 9	syl	$$⊢U\in \mathrm{NrmCVec}\to S:ℂ\times X⟶X$$