Theorem midbtwn

Description: Betweenness 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	$$⊢\phi \to B\in P$$
Assertion	midbtwn	$$⊢\phi \to A{mid}_{𝒢}\left(G\right)B\in \left(AIB\right)$$

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	1 2 3 4 5 6 7	midcl	$$⊢\phi \to A{mid}_{𝒢}\left(G\right)B\in P$$
9		eqid	$$⊢{Line}_{𝒢}\left(G\right)={Line}_{𝒢}\left(G\right)$$
10		eqid	$$⊢{pInv}_{𝒢}\left(G\right)={pInv}_{𝒢}\left(G\right)$$
11		eqid	$$⊢{pInv}_{𝒢}\left(G\right)\left(A{mid}_{𝒢}\left(G\right)B\right)={pInv}_{𝒢}\left(G\right)\left(A{mid}_{𝒢}\left(G\right)B\right)$$
12	1 2 3 9 10 4 8 11 6	mirbtwn	$$⊢\phi \to A{mid}_{𝒢}\left(G\right)B\in \left({pInv}_{𝒢}\left(G\right)\left(A{mid}_{𝒢}\left(G\right)B\right)\left(A\right)IA\right)$$
13		eqidd	$$⊢\phi \to A{mid}_{𝒢}\left(G\right)B=A{mid}_{𝒢}\left(G\right)B$$
14	1 2 3 4 5 6 7 10 8	ismidb	$$⊢\phi \to \left(B={pInv}_{𝒢}\left(G\right)\left(A{mid}_{𝒢}\left(G\right)B\right)\left(A\right)\leftrightarrow A{mid}_{𝒢}\left(G\right)B=A{mid}_{𝒢}\left(G\right)B\right)$$
15	13 14	mpbird	$$⊢\phi \to B={pInv}_{𝒢}\left(G\right)\left(A{mid}_{𝒢}\left(G\right)B\right)\left(A\right)$$
16	15	oveq1d	$$⊢\phi \to BIA={pInv}_{𝒢}\left(G\right)\left(A{mid}_{𝒢}\left(G\right)B\right)\left(A\right)IA$$
17	12 16	eleqtrrd	$$⊢\phi \to A{mid}_{𝒢}\left(G\right)B\in \left(BIA\right)$$
18	1 2 3 4 7 8 6 17	tgbtwncom	$$⊢\phi \to A{mid}_{𝒢}\left(G\right)B\in \left(AIB\right)$$