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 = ( Base ‘ 𝐺 )
	tglngval.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	tglngval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	tglngval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tglngval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
	tglngval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
	tgcolg.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
	ncoltgdim2.1	⊢ ( 𝜑 → ¬ ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ∨ 𝑋 = 𝑌 ) )
Assertion	ncoltgdim2	⊢ ( 𝜑 → 𝐺 DimTarskiG≥ 2 )

Proof

Step	Hyp	Ref	Expression
1		tglngval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tglngval.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
3		tglngval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		tglngval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tglngval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
6		tglngval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
7		tgcolg.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
8		ncoltgdim2.1	⊢ ( 𝜑 → ¬ ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ∨ 𝑋 = 𝑌 ) )
9	4	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝐺 DimTarskiG≥ 2 ) → 𝐺 ∈ TarskiG )
10	5	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝐺 DimTarskiG≥ 2 ) → 𝑋 ∈ 𝑃 )
11	6	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝐺 DimTarskiG≥ 2 ) → 𝑌 ∈ 𝑃 )
12	7	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝐺 DimTarskiG≥ 2 ) → 𝑍 ∈ 𝑃 )
13		simpr	⊢ ( ( 𝜑 ∧ ¬ 𝐺 DimTarskiG≥ 2 ) → ¬ 𝐺 DimTarskiG≥ 2 )
14	1 2 3 9 10 11 12 13	tgdim01ln	⊢ ( ( 𝜑 ∧ ¬ 𝐺 DimTarskiG≥ 2 ) → ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ∨ 𝑋 = 𝑌 ) )
15	8 14	mtand	⊢ ( 𝜑 → ¬ ¬ 𝐺 DimTarskiG≥ 2 )
16	15	notnotrd	⊢ ( 𝜑 → 𝐺 DimTarskiG≥ 2 )