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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 ∈ TopMnd → 𝐹 ∈ ( ( 𝐽 ×_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 ∈ TopMnd ↔ ( 𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ( ( 𝐽 ×_t 𝐽 ) Cn 𝐽 ) ) )
4	3	simp3bi	⊢ ( 𝐺 ∈ TopMnd → 𝐹 ∈ ( ( 𝐽 ×_t 𝐽 ) Cn 𝐽 ) )