Theorem midcom

Description: Commutativity rule for the midpoint. (Contributed by Thierry Arnoux, 2-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	midcom	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) = ( 𝐵 ( midG ‘ 𝐺 ) 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		eqid	⊢ ( LineG ‘ 𝐺 ) = ( LineG ‘ 𝐺 )
9		eqid	⊢ ( pInvG ‘ 𝐺 ) = ( pInvG ‘ 𝐺 )
10	1 2 3 4 5 7 6	midcl	⊢ ( 𝜑 → ( 𝐵 ( midG ‘ 𝐺 ) 𝐴 ) ∈ 𝑃 )
11		eqid	⊢ ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐵 ( midG ‘ 𝐺 ) 𝐴 ) ) = ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐵 ( midG ‘ 𝐺 ) 𝐴 ) )
12		eqidd	⊢ ( 𝜑 → ( 𝐵 ( midG ‘ 𝐺 ) 𝐴 ) = ( 𝐵 ( midG ‘ 𝐺 ) 𝐴 ) )
13	1 2 3 4 5 7 6 12	midcgr	⊢ ( 𝜑 → ( ( 𝐵 ( midG ‘ 𝐺 ) 𝐴 ) − 𝐵 ) = ( ( 𝐵 ( midG ‘ 𝐺 ) 𝐴 ) − 𝐴 ) )
14	1 2 3 4 5 7 6	midbtwn	⊢ ( 𝜑 → ( 𝐵 ( midG ‘ 𝐺 ) 𝐴 ) ∈ ( 𝐵 𝐼 𝐴 ) )
15	1 2 3 8 9 4 10 11 6 7 13 14	ismir	⊢ ( 𝜑 → 𝐵 = ( ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐵 ( midG ‘ 𝐺 ) 𝐴 ) ) ‘ 𝐴 ) )
16	1 2 3 4 5 6 7 9 10	ismidb	⊢ ( 𝜑 → ( 𝐵 = ( ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐵 ( midG ‘ 𝐺 ) 𝐴 ) ) ‘ 𝐴 ) ↔ ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) = ( 𝐵 ( midG ‘ 𝐺 ) 𝐴 ) ) )
17	15 16	mpbid	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) = ( 𝐵 ( midG ‘ 𝐺 ) 𝐴 ) )