Theorem motcl

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

	Ref	Expression
Hypotheses	ismot.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	ismot.m	⊢ − = ( dist ‘ 𝐺 )
	motgrp.1	⊢ ( 𝜑 → 𝐺 ∈ 𝑉 )
	motco.2	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐺 Ismt 𝐺 ) )
	motcl.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
Assertion	motcl	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		ismot.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		ismot.m	⊢ − = ( dist ‘ 𝐺 )
3		motgrp.1	⊢ ( 𝜑 → 𝐺 ∈ 𝑉 )
4		motco.2	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐺 Ismt 𝐺 ) )
5		motcl.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6	1 2 3 4	motf1o	⊢ ( 𝜑 → 𝐹 : 𝑃 –1-1-onto→ 𝑃 )
7		f1of	⊢ ( 𝐹 : 𝑃 –1-1-onto→ 𝑃 → 𝐹 : 𝑃 ⟶ 𝑃 )
8	6 7	syl	⊢ ( 𝜑 → 𝐹 : 𝑃 ⟶ 𝑃 )
9	8 5	ffvelcdmd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ 𝑃 )