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Metamath Proof Explorer

Theorem slmdvsdi

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	slmdvsdi.v	$⊢ V = {Base}_{W}$
		slmdvsdi.a	$⊢ \underset{˙}{+} = +_{W}$
		slmdvsdi.f	$⊢ F = Scalar (W)$
		slmdvsdi.s	$⊢ \underset{˙}{\cdot} = \cdot_{W}$
		slmdvsdi.k	$⊢ K = {Base}_{F}$
	Assertion	slmdvsdi	$⊢ (W \in SLMod \land (R \in K \land X \in V \land Y \in V)) \to R \underset{˙}{\cdot} (X \underset{˙}{+} Y) = (R \underset{˙}{\cdot} X) \underset{˙}{+} (R \underset{˙}{\cdot} Y)$

Proof

Step	Hyp	Ref	Expression
1		slmdvsdi.v	$⊢ V = {Base}_{W}$
2		slmdvsdi.a	$⊢ \underset{˙}{+} = +_{W}$
3		slmdvsdi.f	$⊢ F = Scalar (W)$
4		slmdvsdi.s	$⊢ \underset{˙}{\cdot} = \cdot_{W}$
5		slmdvsdi.k	$⊢ K = {Base}_{F}$
6		eqid	$⊢ 0_{W} = 0_{W}$
7		eqid	$⊢ +_{F} = +_{F}$
8		eqid	$⊢ \cdot_{F} = \cdot_{F}$
9		eqid	$⊢ 1_{F} = 1_{F}$
10		eqid	$⊢ 0_{F} = 0_{F}$
11	1 2 4 6 3 5 7 8 9 10	slmdlema	$⊢ (W \in SLMod \land (R \in K \land R \in K) \land (Y \in V \land X \in V)) \to ((R \underset{˙}{\cdot} X \in V \land R \underset{˙}{\cdot} (X \underset{˙}{+} Y) = (R \underset{˙}{\cdot} X) \underset{˙}{+} (R \underset{˙}{\cdot} Y) \land (R +_{F} R) \underset{˙}{\cdot} X = (R \underset{˙}{\cdot} X) \underset{˙}{+} (R \underset{˙}{\cdot} X)) \land ((R \cdot_{F} R) \underset{˙}{\cdot} X = R \underset{˙}{\cdot} (R \underset{˙}{\cdot} X) \land 1_{F} \underset{˙}{\cdot} X = X \land 0_{F} \underset{˙}{\cdot} X = 0_{W}))$
12	11	simpld	$⊢ (W \in SLMod \land (R \in K \land R \in K) \land (Y \in V \land X \in V)) \to (R \underset{˙}{\cdot} X \in V \land R \underset{˙}{\cdot} (X \underset{˙}{+} Y) = (R \underset{˙}{\cdot} X) \underset{˙}{+} (R \underset{˙}{\cdot} Y) \land (R +_{F} R) \underset{˙}{\cdot} X = (R \underset{˙}{\cdot} X) \underset{˙}{+} (R \underset{˙}{\cdot} X))$
13	12	simp2d	$⊢ (W \in SLMod \land (R \in K \land R \in K) \land (Y \in V \land X \in V)) \to R \underset{˙}{\cdot} (X \underset{˙}{+} Y) = (R \underset{˙}{\cdot} X) \underset{˙}{+} (R \underset{˙}{\cdot} Y)$
14	13	3expia	$⊢ (W \in SLMod \land (R \in K \land R \in K)) \to ((Y \in V \land X \in V) \to R \underset{˙}{\cdot} (X \underset{˙}{+} Y) = (R \underset{˙}{\cdot} X) \underset{˙}{+} (R \underset{˙}{\cdot} Y))$
15	14	anabsan2	$⊢ (W \in SLMod \land R \in K) \to ((Y \in V \land X \in V) \to R \underset{˙}{\cdot} (X \underset{˙}{+} Y) = (R \underset{˙}{\cdot} X) \underset{˙}{+} (R \underset{˙}{\cdot} Y))$
16	15	exp4b	$⊢ W \in SLMod \to (R \in K \to (Y \in V \to (X \in V \to R \underset{˙}{\cdot} (X \underset{˙}{+} Y) = (R \underset{˙}{\cdot} X) \underset{˙}{+} (R \underset{˙}{\cdot} Y))))$
17	16	com34	$⊢ W \in SLMod \to (R \in K \to (X \in V \to (Y \in V \to R \underset{˙}{\cdot} (X \underset{˙}{+} Y) = (R \underset{˙}{\cdot} X) \underset{˙}{+} (R \underset{˙}{\cdot} Y))))$
18	17	3imp2	$⊢ (W \in SLMod \land (R \in K \land X \in V \land Y \in V)) \to R \underset{˙}{\cdot} (X \underset{˙}{+} Y) = (R \underset{˙}{\cdot} X) \underset{˙}{+} (R \underset{˙}{\cdot} Y)$