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Theorem ncoltgdim2 26926

Description: If there are three non-colinear points, then the dimension is at least two. Converse of tglowdim2l 27011. (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 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐺Dim_TarskiG≥2) → 𝐺 ∈ TarskiG)
7		tglngval.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
8	7	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐺Dim_TarskiG≥2) → 𝑋 ∈ 𝑃)
9		tglngval.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
10	9	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐺Dim_TarskiG≥2) → 𝑌 ∈ 𝑃)
11		tgcolg.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
12	11	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐺Dim_TarskiG≥2) → 𝑍 ∈ 𝑃)
13		simpr 485	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐺Dim_TarskiG≥2) → ¬ 𝐺Dim_TarskiG≥2)
14	2, 3, 4, 6, 8, 10, 12, 13	tgdim01ln 26925	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐺Dim_TarskiG≥2) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
15	1, 14	mtand 813	. 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 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 2c2 12028 Basecbs 16912 TarskiGcstrkg 26788 Dim_TarskiG≥cstrkgld 26792 Itvcitv 26794 LineGclng 26795

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-trkgc 26809 df-trkgcb 26811 df-trkgld 26813 df-trkg 26814

This theorem is referenced by: opptgdim2 27106 trgcopy 27165 trgcopyeulem 27166 cgrg3col4 27214

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