Theorem slmdvsdir

Description: Distributive law for scalar product. ( ax-hvdistr1 analog.) (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 22-Sep-2015) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	slmdvsdir.v	$$⊢V={\mathrm{Base}}_W$$
	slmdvsdir.a	$$⊢\underset{\dot{}}{+}={+}_W$$
	slmdvsdir.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
	slmdvsdir.s	$$⊢\underset{\dot{}}{·}={\cdot }_W$$
	slmdvsdir.k	$$⊢K={\mathrm{Base}}_F$$
	slmdvsdir.p	$$⊢\underset{\dot{}}{⨣}={+}_F$$
Assertion	slmdvsdir	$$⊢\left(W\in \mathrm{SLMod}\wedge \left(Q\in K\wedge R\in K\wedge X\in V\right)\right)\to \left(Q\underset{\dot{}}{⨣}R\right)\underset{\dot{}}{·}X=\left(Q\underset{\dot{}}{·}X\right)\underset{\dot{}}{+}\left(R\underset{\dot{}}{·}X\right)$$

Proof

Step	Hyp	Ref	Expression
1		slmdvsdir.v	$$⊢V={\mathrm{Base}}_W$$
2		slmdvsdir.a	$$⊢\underset{\dot{}}{+}={+}_W$$
3		slmdvsdir.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
4		slmdvsdir.s	$$⊢\underset{\dot{}}{·}={\cdot }_W$$
5		slmdvsdir.k	$$⊢K={\mathrm{Base}}_F$$
6		slmdvsdir.p	$$⊢\underset{\dot{}}{⨣}={+}_F$$
7		eqid	$$⊢0_W=0_W$$
8		eqid	$$⊢{\cdot }_F={\cdot }_F$$
9		eqid	$$⊢1_F=1_F$$
10		eqid	$$⊢0_F=0_F$$
11	1 2 4 7 3 5 6 8 9 10	slmdlema	$$⊢\left(W\in \mathrm{SLMod}\wedge \left(Q\in K\wedge R\in K\right)\wedge \left(X\in V\wedge X\in V\right)\right)\to \left(\left(R\underset{\dot{}}{·}X\in V\wedge R\underset{\dot{}}{·}\left(X\underset{\dot{}}{+}X\right)=\left(R\underset{\dot{}}{·}X\right)\underset{\dot{}}{+}\left(R\underset{\dot{}}{·}X\right)\wedge \left(Q\underset{\dot{}}{⨣}R\right)\underset{\dot{}}{·}X=\left(Q\underset{\dot{}}{·}X\right)\underset{\dot{}}{+}\left(R\underset{\dot{}}{·}X\right)\right)\wedge \left(\left(Q{\cdot }_{F}R\right)\underset{\dot{}}{·}X=Q\underset{\dot{}}{·}\left(R\underset{\dot{}}{·}X\right)\wedge 1_F\underset{\dot{}}{·}X=X\wedge 0_F\underset{\dot{}}{·}X=0_W\right)\right)$$
12	11	simpld	$$⊢\left(W\in \mathrm{SLMod}\wedge \left(Q\in K\wedge R\in K\right)\wedge \left(X\in V\wedge X\in V\right)\right)\to \left(R\underset{\dot{}}{·}X\in V\wedge R\underset{\dot{}}{·}\left(X\underset{\dot{}}{+}X\right)=\left(R\underset{\dot{}}{·}X\right)\underset{\dot{}}{+}\left(R\underset{\dot{}}{·}X\right)\wedge \left(Q\underset{\dot{}}{⨣}R\right)\underset{\dot{}}{·}X=\left(Q\underset{\dot{}}{·}X\right)\underset{\dot{}}{+}\left(R\underset{\dot{}}{·}X\right)\right)$$
13	12	simp3d	$$⊢\left(W\in \mathrm{SLMod}\wedge \left(Q\in K\wedge R\in K\right)\wedge \left(X\in V\wedge X\in V\right)\right)\to \left(Q\underset{\dot{}}{⨣}R\right)\underset{\dot{}}{·}X=\left(Q\underset{\dot{}}{·}X\right)\underset{\dot{}}{+}\left(R\underset{\dot{}}{·}X\right)$$
14	13	3expa	$$⊢\left(\left(W\in \mathrm{SLMod}\wedge \left(Q\in K\wedge R\in K\right)\right)\wedge \left(X\in V\wedge X\in V\right)\right)\to \left(Q\underset{\dot{}}{⨣}R\right)\underset{\dot{}}{·}X=\left(Q\underset{\dot{}}{·}X\right)\underset{\dot{}}{+}\left(R\underset{\dot{}}{·}X\right)$$
15	14	anabsan2	$$⊢\left(\left(W\in \mathrm{SLMod}\wedge \left(Q\in K\wedge R\in K\right)\right)\wedge X\in V\right)\to \left(Q\underset{\dot{}}{⨣}R\right)\underset{\dot{}}{·}X=\left(Q\underset{\dot{}}{·}X\right)\underset{\dot{}}{+}\left(R\underset{\dot{}}{·}X\right)$$
16	15	exp42	$$⊢W\in \mathrm{SLMod}\to \left(Q\in K\to \left(R\in K\to \left(X\in V\to \left(Q\underset{\dot{}}{⨣}R\right)\underset{\dot{}}{·}X=\left(Q\underset{\dot{}}{·}X\right)\underset{\dot{}}{+}\left(R\underset{\dot{}}{·}X\right)\right)\right)\right)$$
17	16	3imp2	$$⊢\left(W\in \mathrm{SLMod}\wedge \left(Q\in K\wedge R\in K\wedge X\in V\right)\right)\to \left(Q\underset{\dot{}}{⨣}R\right)\underset{\dot{}}{·}X=\left(Q\underset{\dot{}}{·}X\right)\underset{\dot{}}{+}\left(R\underset{\dot{}}{·}X\right)$$