Theorem midcgr

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

	Ref	Expression
Hypotheses	ismid.p	$$⊢P={\mathrm{Base}}_G$$
	ismid.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	ismid.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	ismid.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	ismid.1	$$⊢\phi \to G{\mathrm{Dim}}_{𝒢}\ge 2$$
	midcl.1	$$⊢\phi \to A\in P$$
	midcl.2	$$⊢\phi \to B\in P$$
	midcgr.1	$$⊢\phi \to A{mid}_{𝒢}\left(G\right)B=C$$
Assertion	midcgr	$$⊢\phi \to C\underset{\dot{}}{-}A=C\underset{\dot{}}{-}B$$

Proof

Step	Hyp	Ref	Expression
1		ismid.p	$$⊢P={\mathrm{Base}}_G$$
2		ismid.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		ismid.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		ismid.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		ismid.1	$$⊢\phi \to G{\mathrm{Dim}}_{𝒢}\ge 2$$
6		midcl.1	$$⊢\phi \to A\in P$$
7		midcl.2	$$⊢\phi \to B\in P$$
8		midcgr.1	$$⊢\phi \to A{mid}_{𝒢}\left(G\right)B=C$$
9		eqid	$$⊢{pInv}_{𝒢}\left(G\right)={pInv}_{𝒢}\left(G\right)$$
10	1 2 3 4 5 6 7	midcl	$$⊢\phi \to A{mid}_{𝒢}\left(G\right)B\in P$$
11	8 10	eqeltrrd	$$⊢\phi \to C\in P$$
12	1 2 3 4 5 6 7 9 11	ismidb	$$⊢\phi \to \left(B={pInv}_{𝒢}\left(G\right)\left(C\right)\left(A\right)\leftrightarrow A{mid}_{𝒢}\left(G\right)B=C\right)$$
13	8 12	mpbird	$$⊢\phi \to B={pInv}_{𝒢}\left(G\right)\left(C\right)\left(A\right)$$
14	13	oveq2d	$$⊢\phi \to C\underset{\dot{}}{-}B=C\underset{\dot{}}{-}{pInv}_{𝒢}\left(G\right)\left(C\right)\left(A\right)$$
15		eqid	$$⊢{Line}_{𝒢}\left(G\right)={Line}_{𝒢}\left(G\right)$$
16		eqid	$$⊢{pInv}_{𝒢}\left(G\right)\left(C\right)={pInv}_{𝒢}\left(G\right)\left(C\right)$$
17	1 2 3 15 9 4 11 16 6	mircgr	$$⊢\phi \to C\underset{\dot{}}{-}{pInv}_{𝒢}\left(G\right)\left(C\right)\left(A\right)=C\underset{\dot{}}{-}A$$
18	14 17	eqtr2d	$$⊢\phi \to C\underset{\dot{}}{-}A=C\underset{\dot{}}{-}B$$