Theorem midid

Description: Midpoint of a null segment. (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	midid	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐴 ) = 𝐴 )

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 6	midcl	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐴 ) ∈ 𝑃 )
9	1 2 3 4 5 6 6	midbtwn	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐴 ) ∈ ( 𝐴 𝐼 𝐴 ) )
10	1 2 3 4 6 8 9	axtgbtwnid	⊢ ( 𝜑 → 𝐴 = ( 𝐴 ( midG ‘ 𝐺 ) 𝐴 ) )
11	10	eqcomd	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐴 ) = 𝐴 )