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Metamath Proof Explorer

Theorem slmdvsass

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

		Ref	Expression
	Hypotheses	slmdvsass.v	$⊢ V = {Base}_{W}$
		slmdvsass.f	$⊢ F = Scalar (W)$
		slmdvsass.s	$⊢ \underset{˙}{\cdot} = \cdot_{W}$
		slmdvsass.k	$⊢ K = {Base}_{F}$
		slmdvsass.t	$⊢ \underset{˙}{\times} = \cdot_{F}$
	Assertion	slmdvsass	$⊢ (W \in SLMod \land (Q \in K \land R \in K \land X \in V)) \to (Q \underset{˙}{\times} R) \underset{˙}{\cdot} X = Q \underset{˙}{\cdot} (R \underset{˙}{\cdot} X)$

Proof

Step	Hyp	Ref	Expression
1		slmdvsass.v	$⊢ V = {Base}_{W}$
2		slmdvsass.f	$⊢ F = Scalar (W)$
3		slmdvsass.s	$⊢ \underset{˙}{\cdot} = \cdot_{W}$
4		slmdvsass.k	$⊢ K = {Base}_{F}$
5		slmdvsass.t	$⊢ \underset{˙}{\times} = \cdot_{F}$
6		eqid	$⊢ +_{W} = +_{W}$
7		eqid	$⊢ 0_{W} = 0_{W}$
8		eqid	$⊢ +_{F} = +_{F}$
9		eqid	$⊢ 1_{F} = 1_{F}$
10		eqid	$⊢ 0_{F} = 0_{F}$
11	1 6 3 7 2 4 8 5 9 10	slmdlema	$⊢ (W \in SLMod \land (Q \in K \land R \in K) \land (X \in V \land X \in V)) \to ((R \underset{˙}{\cdot} X \in V \land R \underset{˙}{\cdot} (X +_{W} X) = (R \underset{˙}{\cdot} X) +_{W} (R \underset{˙}{\cdot} X) \land (Q +_{F} R) \underset{˙}{\cdot} X = (Q \underset{˙}{\cdot} X) +_{W} (R \underset{˙}{\cdot} X)) \land ((Q \underset{˙}{\times} R) \underset{˙}{\cdot} X = Q \underset{˙}{\cdot} (R \underset{˙}{\cdot} X) \land 1_{F} \underset{˙}{\cdot} X = X \land 0_{F} \underset{˙}{\cdot} X = 0_{W}))$
12	11	simprd	$⊢ (W \in SLMod \land (Q \in K \land R \in K) \land (X \in V \land X \in V)) \to ((Q \underset{˙}{\times} R) \underset{˙}{\cdot} X = Q \underset{˙}{\cdot} (R \underset{˙}{\cdot} X) \land 1_{F} \underset{˙}{\cdot} X = X \land 0_{F} \underset{˙}{\cdot} X = 0_{W})$
13	12	simp1d	$⊢ (W \in SLMod \land (Q \in K \land R \in K) \land (X \in V \land X \in V)) \to (Q \underset{˙}{\times} R) \underset{˙}{\cdot} X = Q \underset{˙}{\cdot} (R \underset{˙}{\cdot} X)$
14	13	3expa	$⊢ ((W \in SLMod \land (Q \in K \land R \in K)) \land (X \in V \land X \in V)) \to (Q \underset{˙}{\times} R) \underset{˙}{\cdot} X = Q \underset{˙}{\cdot} (R \underset{˙}{\cdot} X)$
15	14	anabsan2	$⊢ ((W \in SLMod \land (Q \in K \land R \in K)) \land X \in V) \to (Q \underset{˙}{\times} R) \underset{˙}{\cdot} X = Q \underset{˙}{\cdot} (R \underset{˙}{\cdot} X)$
16	15	exp42	$⊢ W \in SLMod \to (Q \in K \to (R \in K \to (X \in V \to (Q \underset{˙}{\times} R) \underset{˙}{\cdot} X = Q \underset{˙}{\cdot} (R \underset{˙}{\cdot} X))))$
17	16	3imp2	$⊢ (W \in SLMod \land (Q \in K \land R \in K \land X \in V)) \to (Q \underset{˙}{\times} R) \underset{˙}{\cdot} X = Q \underset{˙}{\cdot} (R \underset{˙}{\cdot} X)$