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	$$⊢J=\mathrm{TopOpen}\left(G\right)$$
	tgpcn.1	$$⊢F={+}_{𝑓}\left(G\right)$$
Assertion	tmdcn	$$⊢G\in \mathrm{TopMnd}\to F\in \left(\left(J{\times }_{t}J\right)\mathrm{Cn}J\right)$$

Proof

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