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Theorem ncoltgdim2 28655

Description: If there are three non-colinear points, then the dimension is at least two. Converse of tglowdim2l 28740. (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 28654	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐺Dim_TarskiG≥2) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
15	1, 14	mtand 816	. 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 848 = wceq 1542 ∈ wcel 2114 class class class wbr 5100 ‘cfv 6502 (class class class)co 7370 2c2 12214 Basecbs 17150 TarskiGcstrkg 28516 Dim_TarskiG≥cstrkgld 28520 Itvcitv 28522 LineGclng 28523

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-om 7821 df-1st 7945 df-2nd 7946 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-er 8647 df-en 8898 df-dom 8899 df-sdom 8900 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-nn 12160 df-2 12222 df-n0 12416 df-z 12503 df-uz 12766 df-fz 13438 df-fzo 13585 df-trkgc 28537 df-trkgcb 28539 df-trkgld 28541 df-trkg 28542

This theorem is referenced by: opptgdim2 28835 trgcopy 28894 trgcopyeulem 28895 cgrg3col4 28943

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