Theorem motcl

Description: Closure of motions. (Contributed by Thierry Arnoux, 15-Dec-2019)

	Ref	Expression
Hypotheses	ismot.p	$$⊢P={\mathrm{Base}}_G$$
	ismot.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	motgrp.1	$$⊢\phi \to G\in V$$
	motco.2	$$⊢\phi \to F\in \left(GIsmtG\right)$$
	motcl.a	$$⊢\phi \to A\in P$$
Assertion	motcl	$$⊢\phi \to F\left(A\right)\in P$$

Proof

Step	Hyp	Ref	Expression
1		ismot.p	$$⊢P={\mathrm{Base}}_G$$
2		ismot.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		motgrp.1	$$⊢\phi \to G\in V$$
4		motco.2	$$⊢\phi \to F\in \left(GIsmtG\right)$$
5		motcl.a	$$⊢\phi \to A\in P$$
6	1 2 3 4	motf1o	$$⊢\phi \to F:P\underset{1-1\; onto}{⟶}P$$
7		f1of	$$⊢F:P\underset{1-1\; onto}{⟶}P\to F:P⟶P$$
8	6 7	syl	$$⊢\phi \to F:P⟶P$$
9	8 5	ffvelcdmd	$$⊢\phi \to F\left(A\right)\in P$$