Theorem midcl

Description: Closure of the midpoint. (Contributed by Thierry Arnoux, 1-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	midcl	$$⊢\phi \to A{mid}_{𝒢}\left(G\right)B\in P$$

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	midf	$$⊢\phi \to {mid}_{𝒢}\left(G\right):P\times P⟶P$$
9	8 6 7	fovrnd	$$⊢\phi \to A{mid}_{𝒢}\left(G\right)B\in P$$