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	$$⊢F=\left(x\in X⟼A\underset{\dot{}}{+}x\right)$$
	tgplacthmeo.2	$$⊢X={\mathrm{Base}}_G$$
	tgplacthmeo.3	$$⊢\underset{\dot{}}{+}={+}_G$$
	tgplacthmeo.4	$$⊢J=\mathrm{TopOpen}\left(G\right)$$
Assertion	tmdlactcn	$$⊢\left(G\in \mathrm{TopMnd}\wedge A\in X\right)\to F\in \left(J\mathrm{Cn}J\right)$$

Proof

Step	Hyp	Ref	Expression
1		tgplacthmeo.1	$$⊢F=\left(x\in X⟼A\underset{\dot{}}{+}x\right)$$
2		tgplacthmeo.2	$$⊢X={\mathrm{Base}}_G$$
3		tgplacthmeo.3	$$⊢\underset{\dot{}}{+}={+}_G$$
4		tgplacthmeo.4	$$⊢J=\mathrm{TopOpen}\left(G\right)$$
5		simpl	$$⊢\left(G\in \mathrm{TopMnd}\wedge A\in X\right)\to G\in \mathrm{TopMnd}$$
6	4 2	tmdtopon	$$⊢G\in \mathrm{TopMnd}\to J\in \mathrm{TopOn}\left(X\right)$$
7	6	adantr	$$⊢\left(G\in \mathrm{TopMnd}\wedge A\in X\right)\to J\in \mathrm{TopOn}\left(X\right)$$
8		simpr	$$⊢\left(G\in \mathrm{TopMnd}\wedge A\in X\right)\to A\in X$$
9	7 7 8	cnmptc	$$⊢\left(G\in \mathrm{TopMnd}\wedge A\in X\right)\to \left(x\in X⟼A\right)\in \left(J\mathrm{Cn}J\right)$$
10	7	cnmptid	$$⊢\left(G\in \mathrm{TopMnd}\wedge A\in X\right)\to \left(x\in X⟼x\right)\in \left(J\mathrm{Cn}J\right)$$
11	4 3 5 7 9 10	cnmpt1plusg	$$⊢\left(G\in \mathrm{TopMnd}\wedge A\in X\right)\to \left(x\in X⟼A\underset{\dot{}}{+}x\right)\in \left(J\mathrm{Cn}J\right)$$
12	1 11	eqeltrid	$$⊢\left(G\in \mathrm{TopMnd}\wedge A\in X\right)\to F\in \left(J\mathrm{Cn}J\right)$$