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Metamath Proof Explorer

Theorem cnmpt1mulr

Description: Continuity of ring multiplication; analogue of cnmpt12f which cannot be used directly because .r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015)

		Ref	Expression
	Hypotheses	mulrcn.j	$⊢ J = TopOpen (R)$
		cnmpt1mulr.t	$⊢ \underset{˙}{\cdot} = \cdot_{R}$
		cnmpt1mulr.r	$⊢ φ \to R \in TopRing$
		cnmpt1mulr.k	$⊢ φ \to K \in TopOn (X)$
		cnmpt1mulr.a	$⊢ φ \to (x \in X ⟼ A) \in (K Cn J)$
		cnmpt1mulr.b	$⊢ φ \to (x \in X ⟼ B) \in (K Cn J)$
	Assertion	cnmpt1mulr	$⊢ φ \to (x \in X ⟼ A \underset{˙}{\cdot} B) \in (K Cn J)$

Proof

Step	Hyp	Ref	Expression
1		mulrcn.j	$⊢ J = TopOpen (R)$
2		cnmpt1mulr.t	$⊢ \underset{˙}{\cdot} = \cdot_{R}$
3		cnmpt1mulr.r	$⊢ φ \to R \in TopRing$
4		cnmpt1mulr.k	$⊢ φ \to K \in TopOn (X)$
5		cnmpt1mulr.a	$⊢ φ \to (x \in X ⟼ A) \in (K Cn J)$
6		cnmpt1mulr.b	$⊢ φ \to (x \in X ⟼ B) \in (K Cn J)$
7		eqid	$⊢ {mulGrp}_{R} = {mulGrp}_{R}$
8	7 1	mgptopn	$⊢ J = TopOpen ({mulGrp}_{R})$
9	7 2	mgpplusg	$⊢ \underset{˙}{\cdot} = +_{{mulGrp}_{R}}$
10	7	trgtmd	$⊢ R \in TopRing \to {mulGrp}_{R} \in TopMnd$
11	3 10	syl	$⊢ φ \to {mulGrp}_{R} \in TopMnd$
12	8 9 11 4 5 6	cnmpt1plusg	$⊢ φ \to (x \in X ⟼ A \underset{˙}{\cdot} B) \in (K Cn J)$