Theorem midcgr

Description: Congruence 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 )
	midcgr.1	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) = 𝐶 )
Assertion	midcgr	⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = ( 𝐶 − 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		midcgr.1	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) = 𝐶 )
9		eqid	⊢ ( pInvG ‘ 𝐺 ) = ( pInvG ‘ 𝐺 )
10	1 2 3 4 5 6 7	midcl	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) ∈ 𝑃 )
11	8 10	eqeltrrd	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
12	1 2 3 4 5 6 7 9 11	ismidb	⊢ ( 𝜑 → ( 𝐵 = ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐶 ) ‘ 𝐴 ) ↔ ( 𝐴 ( midG ‘ 𝐺 ) 𝐵 ) = 𝐶 ) )
13	8 12	mpbird	⊢ ( 𝜑 → 𝐵 = ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐶 ) ‘ 𝐴 ) )
14	13	oveq2d	⊢ ( 𝜑 → ( 𝐶 − 𝐵 ) = ( 𝐶 − ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐶 ) ‘ 𝐴 ) ) )
15		eqid	⊢ ( LineG ‘ 𝐺 ) = ( LineG ‘ 𝐺 )
16		eqid	⊢ ( ( pInvG ‘ 𝐺 ) ‘ 𝐶 ) = ( ( pInvG ‘ 𝐺 ) ‘ 𝐶 )
17	1 2 3 15 9 4 11 16 6	mircgr	⊢ ( 𝜑 → ( 𝐶 − ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐶 ) ‘ 𝐴 ) ) = ( 𝐶 − 𝐴 ) )
18	14 17	eqtr2d	⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = ( 𝐶 − 𝐵 ) )