Theorem mirmir

Description: The point inversion function is an involution. Theorem 7.7 of Schwabhauser p. 49. (Contributed by Thierry Arnoux, 3-Jun-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 )
	mirmir.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
Assertion	mirmir	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑀 ‘ 𝐵 ) ) = 𝐵 )

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		mirmir.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
10	1 2 3 4 5 6 7 8 9	mircl	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) ∈ 𝑃 )
11	1 2 3 4 5 6 7 8 9	mircgr	⊢ ( 𝜑 → ( 𝐴 − ( 𝑀 ‘ 𝐵 ) ) = ( 𝐴 − 𝐵 ) )
12	11	eqcomd	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐴 − ( 𝑀 ‘ 𝐵 ) ) )
13	1 2 3 4 5 6 7 8 9	mirbtwn	⊢ ( 𝜑 → 𝐴 ∈ ( ( 𝑀 ‘ 𝐵 ) 𝐼 𝐵 ) )
14	1 2 3 6 10 7 9 13	tgbtwncom	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 𝐼 ( 𝑀 ‘ 𝐵 ) ) )
15	1 2 3 4 5 6 7 8 10 9 12 14	ismir	⊢ ( 𝜑 → 𝐵 = ( 𝑀 ‘ ( 𝑀 ‘ 𝐵 ) ) )
16	15	eqcomd	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑀 ‘ 𝐵 ) ) = 𝐵 )