Theorem nv0

Description: Zero times a vector is the zero vector. (Contributed by NM, 27-Nov-2007) (Revised by Mario Carneiro, 21-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nv0.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	nv0.4	$$⊢S={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
	nv0.6	$$⊢Z=0_{\mathrm{vec}}\left(U\right)$$
Assertion	nv0	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\right)\to 0SA=Z$$

Proof

Step	Hyp	Ref	Expression
1		nv0.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		nv0.4	$$⊢S={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
3		nv0.6	$$⊢Z=0_{\mathrm{vec}}\left(U\right)$$
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		eqid	$$⊢{+}_v\left(U\right)={+}_v\left(U\right)$$
7	6	vafval	$$⊢{+}_v\left(U\right)=1^{st}\left(1^{st}\left(U\right)\right)$$
8	2	smfval	$$⊢S=2^{nd}\left(1^{st}\left(U\right)\right)$$
9	1 6	bafval	$$⊢X=\mathrm{ran}{+}_v\left(U\right)$$
10		eqid	$$⊢\mathrm{GId}\left({+}_v\left(U\right)\right)=\mathrm{GId}\left({+}_v\left(U\right)\right)$$
11	7 8 9 10	vc0	$$⊢\left(1^{st}\left(U\right)\in {CVec}_{\mathrm{OLD}}\wedge A\in X\right)\to 0SA=\mathrm{GId}\left({+}_v\left(U\right)\right)$$
12	5 11	sylan	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\right)\to 0SA=\mathrm{GId}\left({+}_v\left(U\right)\right)$$
13	6 3	0vfval	$$⊢U\in \mathrm{NrmCVec}\to Z=\mathrm{GId}\left({+}_v\left(U\right)\right)$$
14	13	adantr	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\right)\to Z=\mathrm{GId}\left({+}_v\left(U\right)\right)$$
15	12 14	eqtr4d	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\right)\to 0SA=Z$$