Theorem mircl

Description: Closure of the point inversion function. (Contributed by Thierry Arnoux, 20-Oct-2019)

	Ref	Expression
Hypotheses	mirval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	mirval.d	⊢ − = ( dist ‘ 𝐺 )
	mirval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	mirval.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	mirval.s	⊢ 𝑆 = ( pInvG ‘ 𝐺 )
	mirval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	mirval.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	mirfv.m	⊢ 𝑀 = ( 𝑆 ‘ 𝐴 )
	mircl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
Assertion	mircl	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		mirval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		mirval.d	⊢ − = ( dist ‘ 𝐺 )
3		mirval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		mirval.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
5		mirval.s	⊢ 𝑆 = ( pInvG ‘ 𝐺 )
6		mirval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
7		mirval.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
8		mirfv.m	⊢ 𝑀 = ( 𝑆 ‘ 𝐴 )
9		mircl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
10	1 2 3 4 5 6 7 8	mirf	⊢ ( 𝜑 → 𝑀 : 𝑃 ⟶ 𝑃 )
11	10 9	ffvelcdmd	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ 𝑃 )