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Metamath Proof Explorer

Theorem tmdcn

Description: In a topological monoid, the operation F representing the functionalization of the operator slot +g is continuous. (Contributed by Mario Carneiro, 19-Sep-2015)

	Ref	Expression
Hypotheses	tgpcn.j	$⊢ J = TopOpen (G)$
	tgpcn.1	$⊢ F = +_{𝑓} (G)$
Assertion	tmdcn	$⊢ G \in TopMnd \to F \in ((J \times_{t} J) Cn J)$

Proof

Step	Hyp	Ref	Expression
1		tgpcn.j	$⊢ J = TopOpen (G)$
2		tgpcn.1	$⊢ F = +_{𝑓} (G)$
3	2 1	istmd	$⊢ G \in TopMnd \leftrightarrow (G \in Mnd \land G \in TopSp \land F \in ((J \times_{t} J) Cn J))$
4	3	simp3bi	$⊢ G \in TopMnd \to F \in ((J \times_{t} J) Cn J)$