Theorem tmdlactcn

Description: The left group action of element A in a topological monoid G is a continuous function. (Contributed by FL, 18-Mar-2008) (Revised by Mario Carneiro, 14-Aug-2015)

	Ref	Expression
Hypotheses	tgplacthmeo.1	⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 + 𝑥 ) )
	tgplacthmeo.2	⊢ 𝑋 = ( Base ‘ 𝐺 )
	tgplacthmeo.3	⊢ + = ( +g ‘ 𝐺 )
	tgplacthmeo.4	⊢ 𝐽 = ( TopOpen ‘ 𝐺 )
Assertion	tmdlactcn	⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn 𝐽 ) )

Proof

Step	Hyp	Ref	Expression
1		tgplacthmeo.1	⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 + 𝑥 ) )
2		tgplacthmeo.2	⊢ 𝑋 = ( Base ‘ 𝐺 )
3		tgplacthmeo.3	⊢ + = ( +g ‘ 𝐺 )
4		tgplacthmeo.4	⊢ 𝐽 = ( TopOpen ‘ 𝐺 )
5		simpl	⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → 𝐺 ∈ TopMnd )
6	4 2	tmdtopon	⊢ ( 𝐺 ∈ TopMnd → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
7	6	adantr	⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
8		simpr	⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → 𝐴 ∈ 𝑋 )
9	7 7 8	cnmptc	⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ∈ ( 𝐽 Cn 𝐽 ) )
10	7	cnmptid	⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → ( 𝑥 ∈ 𝑋 ↦ 𝑥 ) ∈ ( 𝐽 Cn 𝐽 ) )
11	4 3 5 7 9 10	cnmpt1plusg	⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 + 𝑥 ) ) ∈ ( 𝐽 Cn 𝐽 ) )
12	1 11	eqeltrid	⊢ ( ( 𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn 𝐽 ) )