Theorem nvdir

Description: Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007) (New usage is discouraged.)

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

Proof

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