Metamath Proof Explorer

Theorem lmodvscl

Description: Closure of scalar product for a left module. ( hvmulcl analog.) (Contributed by NM, 8-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014)

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

Proof

Step	Hyp	Ref	Expression
1		lmodvscl.v	$⊢ V = {Base}_{W}$
2		lmodvscl.f	$⊢ F = Scalar (W)$
3		lmodvscl.s	$⊢ \underset{˙}{\cdot} = \cdot_{W}$
4		lmodvscl.k	$⊢ K = {Base}_{F}$
5		biid	$⊢ W \in LMod \leftrightarrow W \in LMod$
6		pm4.24	$⊢ R \in K \leftrightarrow (R \in K \land R \in K)$
7		pm4.24	$⊢ X \in V \leftrightarrow (X \in V \land X \in V)$
8		eqid	$⊢ +_{W} = +_{W}$
9		eqid	$⊢ +_{F} = +_{F}$
10		eqid	$⊢ \cdot_{F} = \cdot_{F}$
11		eqid	$⊢ 1_{F} = 1_{F}$
12	1 8 3 2 4 9 10 11	lmodlema	$⊢ (W \in LMod \land (R \in K \land R \in K) \land (X \in V \land X \in V)) \to ((R \underset{˙}{\cdot} X \in V \land R \underset{˙}{\cdot} (X +_{W} X) = (R \underset{˙}{\cdot} X) +_{W} (R \underset{˙}{\cdot} X) \land (R +_{F} R) \underset{˙}{\cdot} X = (R \underset{˙}{\cdot} X) +_{W} (R \underset{˙}{\cdot} X)) \land ((R \cdot_{F} R) \underset{˙}{\cdot} X = R \underset{˙}{\cdot} (R \underset{˙}{\cdot} X) \land 1_{F} \underset{˙}{\cdot} X = X))$
13	12	simpld	$⊢ (W \in LMod \land (R \in K \land R \in K) \land (X \in V \land X \in V)) \to (R \underset{˙}{\cdot} X \in V \land R \underset{˙}{\cdot} (X +_{W} X) = (R \underset{˙}{\cdot} X) +_{W} (R \underset{˙}{\cdot} X) \land (R +_{F} R) \underset{˙}{\cdot} X = (R \underset{˙}{\cdot} X) +_{W} (R \underset{˙}{\cdot} X))$
14	13	simp1d	$⊢ (W \in LMod \land (R \in K \land R \in K) \land (X \in V \land X \in V)) \to R \underset{˙}{\cdot} X \in V$
15	5 6 7 14	syl3anb	$⊢ (W \in LMod \land R \in K \land X \in V) \to R \underset{˙}{\cdot} X \in V$