Theorem midbtwn

Description: Betweenness of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019)

	Ref	Expression
Hypotheses	ismid.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	ismid.d	⊢ − = ( dist ‘ 𝐺 )
	ismid.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	ismid.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	ismid.1	⊢ ( 𝜑 → 𝐺 DimTarskiG≥ 2 )
	midcl.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	midcl.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
Assertion	midbtwn	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) ∈ ( 𝐴 𝐼 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ismid.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		ismid.d	⊢ − = ( dist ‘ 𝐺 )
3		ismid.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		ismid.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		ismid.1	⊢ ( 𝜑 → 𝐺 DimTarskiG≥ 2 )
6		midcl.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
7		midcl.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
8	1 2 3 4 5 6 7	midcl	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) ∈ 𝑃 )
9		eqid	⊢ ( LineG ‘ 𝐺 ) = ( LineG ‘ 𝐺 )
10		eqid	⊢ ( pInvG ‘ 𝐺 ) = ( pInvG ‘ 𝐺 )
11		eqid	⊢ ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) ) = ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) )
12	1 2 3 9 10 4 8 11 6	mirbtwn	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) ∈ ( ( ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) ) ‘ 𝐴 ) 𝐼 𝐴 ) )
13		eqidd	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) = ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) )
14	1 2 3 4 5 6 7 10 8	ismidb	⊢ ( 𝜑 → ( 𝐵 = ( ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) ) ‘ 𝐴 ) ↔ ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) = ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) ) )
15	13 14	mpbird	⊢ ( 𝜑 → 𝐵 = ( ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) ) ‘ 𝐴 ) )
16	15	oveq1d	⊢ ( 𝜑 → ( 𝐵 𝐼 𝐴 ) = ( ( ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) ) ‘ 𝐴 ) 𝐼 𝐴 ) )
17	12 16	eleqtrrd	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) ∈ ( 𝐵 𝐼 𝐴 ) )
18	1 2 3 4 7 8 6 17	tgbtwncom	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) ∈ ( 𝐴 𝐼 𝐵 ) )