Theorem ncoltgdim2

Description: If there are three non-colinear points, then the dimension is at least two. Converse of tglowdim2l . (Contributed by Thierry Arnoux, 23-Feb-2020)

	Ref	Expression
Hypotheses	tglngval.p	$$⊢P={\mathrm{Base}}_G$$
	tglngval.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
	tglngval.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	tglngval.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tglngval.x	$$⊢\phi \to X\in P$$
	tglngval.y	$$⊢\phi \to Y\in P$$
	tgcolg.z	$$⊢\phi \to Z\in P$$
	ncoltgdim2.1	$$⊢\phi \to \neg \left(Z\in \left(XLY\right)\vee X=Y\right)$$
Assertion	ncoltgdim2	$$⊢\phi \to G{\mathrm{Dim}}_{𝒢}\ge 2$$

Proof

Step	Hyp	Ref	Expression
1		tglngval.p	$$⊢P={\mathrm{Base}}_G$$
2		tglngval.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
3		tglngval.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		tglngval.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		tglngval.x	$$⊢\phi \to X\in P$$
6		tglngval.y	$$⊢\phi \to Y\in P$$
7		tgcolg.z	$$⊢\phi \to Z\in P$$
8		ncoltgdim2.1	$$⊢\phi \to \neg \left(Z\in \left(XLY\right)\vee X=Y\right)$$
9	4	adantr	$$⊢\left(\phi \wedge \neg G{\mathrm{Dim}}_{𝒢}\ge 2\right)\to G\in {𝒢}_{\mathrm{Tarski}}$$
10	5	adantr	$$⊢\left(\phi \wedge \neg G{\mathrm{Dim}}_{𝒢}\ge 2\right)\to X\in P$$
11	6	adantr	$$⊢\left(\phi \wedge \neg G{\mathrm{Dim}}_{𝒢}\ge 2\right)\to Y\in P$$
12	7	adantr	$$⊢\left(\phi \wedge \neg G{\mathrm{Dim}}_{𝒢}\ge 2\right)\to Z\in P$$
13		simpr	$$⊢\left(\phi \wedge \neg G{\mathrm{Dim}}_{𝒢}\ge 2\right)\to \neg G{\mathrm{Dim}}_{𝒢}\ge 2$$
14	1 2 3 9 10 11 12 13	tgdim01ln	$$⊢\left(\phi \wedge \neg G{\mathrm{Dim}}_{𝒢}\ge 2\right)\to \left(Z\in \left(XLY\right)\vee X=Y\right)$$
15	8 14	mtand	$$⊢\phi \to \neg \neg G{\mathrm{Dim}}_{𝒢}\ge 2$$
16	15	notnotrd	$$⊢\phi \to G{\mathrm{Dim}}_{𝒢}\ge 2$$