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Theorem ncoltgdim2 26830

Description: If there are three non-colinear points, then the dimension is at least two. Converse of tglowdim2l 26915. (Contributed by Thierry Arnoux, 23-Feb-2020.)

**Hypotheses**
Ref	Expression
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 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))

**Assertion**
Ref	Expression
ncoltgdim2	⊢ (𝜑 → 𝐺Dim_TarskiG≥2)

**Proof of Theorem ncoltgdim2**
Step	Hyp	Ref	Expression
1		ncoltgdim2.1	. . 3 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2		tglngval.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
3		tglngval.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
4		tglngval.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
5		tglngval.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐺Dim_TarskiG≥2) → 𝐺 ∈ TarskiG)
7		tglngval.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
8	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐺Dim_TarskiG≥2) → 𝑋 ∈ 𝑃)
9		tglngval.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐺Dim_TarskiG≥2) → 𝑌 ∈ 𝑃)
11		tgcolg.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
12	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐺Dim_TarskiG≥2) → 𝑍 ∈ 𝑃)
13		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐺Dim_TarskiG≥2) → ¬ 𝐺Dim_TarskiG≥2)
14	2, 3, 4, 6, 8, 10, 12, 13	tgdim01ln 26829	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐺Dim_TarskiG≥2) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
15	1, 14	mtand 812	. 2 ⊢ (𝜑 → ¬ ¬ 𝐺Dim_TarskiG≥2)
16	15	notnotrd 133	1 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 2c2 11958 Basecbs 16840 TarskiGcstrkg 26693 Dim_TarskiG≥cstrkgld 26697 Itvcitv 26699 LineGclng 26700

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312 df-trkgc 26713 df-trkgcb 26715 df-trkgld 26717 df-trkg 26718

This theorem is referenced by: opptgdim2 27010 trgcopy 27069 trgcopyeulem 27070 cgrg3col4 27118

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