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Mirrors > Home > MPE Home > Th. List > tgdim01ln | Structured version Visualization version GIF version |
Description: In geometries of dimension less than two, then any three points are colinear. (Contributed by Thierry Arnoux, 27-Aug-2019.) |
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) |
tgdim01ln.1 | ⊢ (𝜑 → ¬ 𝐺DimTarskiG≥2) |
Ref | Expression |
---|---|
tgdim01ln | ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglngval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tglngval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
3 | tglngval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tglngval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐺 ∈ TarskiG) |
6 | tglngval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ 𝑃) |
8 | tglngval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
9 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ 𝑃) |
10 | tgcolg.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
11 | 10 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ 𝑃) |
12 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ (𝑋𝐼𝑌)) | |
13 | 1, 2, 3, 5, 7, 9, 11, 12 | btwncolg1 26269 | . 2 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
14 | 4 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝐺 ∈ TarskiG) |
15 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝑋 ∈ 𝑃) |
16 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝑌 ∈ 𝑃) |
17 | 10 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝑍 ∈ 𝑃) |
18 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝑋 ∈ (𝑍𝐼𝑌)) | |
19 | 1, 2, 3, 14, 15, 16, 17, 18 | btwncolg2 26270 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
20 | 4 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐺 ∈ TarskiG) |
21 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑋 ∈ 𝑃) |
22 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ 𝑃) |
23 | 10 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑍 ∈ 𝑃) |
24 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ (𝑋𝐼𝑍)) | |
25 | 1, 2, 3, 20, 21, 22, 23, 24 | btwncolg3 26271 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
26 | tgdim01ln.1 | . . 3 ⊢ (𝜑 → ¬ 𝐺DimTarskiG≥2) | |
27 | 1, 3, 4, 26, 6, 8, 10 | tgdim01 26221 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) |
28 | 13, 19, 25, 27 | mpjao3dan 1423 | 1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 2c2 11681 Basecbs 16473 TarskiGcstrkg 26144 DimTarskiG≥cstrkgld 26148 Itvcitv 26150 LineGclng 26151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-n0 11887 df-z 11971 df-uz 12233 df-fz 12883 df-fzo 13024 df-trkgc 26162 df-trkgcb 26164 df-trkgld 26166 df-trkg 26167 |
This theorem is referenced by: ncoltgdim2 26279 |
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