Theorem mirbtwni

Description: Point inversion preserves betweenness, first half of Theorem 7.15 of Schwabhauser p. 51. (Contributed by Thierry Arnoux, 9-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 )
	miriso.1	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
	miriso.2	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
	mirbtwni.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
	mirbtwni.b	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) )
Assertion	mirbtwni	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑌 ) ∈ ( ( 𝑀 ‘ 𝑋 ) 𝐼 ( 𝑀 ‘ 𝑍 ) ) )

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		miriso.1	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
10		miriso.2	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
11		mirbtwni.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
12		mirbtwni.b	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) )
13		eqid	⊢ ( cgrG ‘ 𝐺 ) = ( cgrG ‘ 𝐺 )
14	1 2 3 4 5 6 7 8	mirf	⊢ ( 𝜑 → 𝑀 : 𝑃 ⟶ 𝑃 )
15	14 9	ffvelrnd	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ 𝑃 )
16	14 10	ffvelrnd	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑌 ) ∈ 𝑃 )
17	14 11	ffvelrnd	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑍 ) ∈ 𝑃 )
18	1 2 3 4 5 6 7 8 9 10	miriso	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) − ( 𝑀 ‘ 𝑌 ) ) = ( 𝑋 − 𝑌 ) )
19	18	eqcomd	⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) = ( ( 𝑀 ‘ 𝑋 ) − ( 𝑀 ‘ 𝑌 ) ) )
20	1 2 3 4 5 6 7 8 10 11	miriso	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑌 ) − ( 𝑀 ‘ 𝑍 ) ) = ( 𝑌 − 𝑍 ) )
21	20	eqcomd	⊢ ( 𝜑 → ( 𝑌 − 𝑍 ) = ( ( 𝑀 ‘ 𝑌 ) − ( 𝑀 ‘ 𝑍 ) ) )
22	1 2 3 4 5 6 7 8 11 9	miriso	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑍 ) − ( 𝑀 ‘ 𝑋 ) ) = ( 𝑍 − 𝑋 ) )
23	22	eqcomd	⊢ ( 𝜑 → ( 𝑍 − 𝑋 ) = ( ( 𝑀 ‘ 𝑍 ) − ( 𝑀 ‘ 𝑋 ) ) )
24	1 2 13 6 9 10 11 15 16 17 19 21 23	trgcgr	⊢ ( 𝜑 → ⟨“ 𝑋 𝑌 𝑍 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ ( 𝑀 ‘ 𝑋 ) ( 𝑀 ‘ 𝑌 ) ( 𝑀 ‘ 𝑍 ) ”⟩ )
25	1 2 3 13 6 9 10 11 15 16 17 24 12	tgbtwnxfr	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑌 ) ∈ ( ( 𝑀 ‘ 𝑋 ) 𝐼 ( 𝑀 ‘ 𝑍 ) ) )