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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 483 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐺 ∈ TarskiG) |
6 | tglngval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ 𝑃) |
8 | tglngval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
9 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ 𝑃) |
10 | tgcolg.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
11 | 10 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ 𝑃) |
12 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ (𝑋𝐼𝑌)) | |
13 | 1, 2, 3, 5, 7, 9, 11, 12 | btwncolg1 26333 | . 2 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
14 | 4 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝐺 ∈ TarskiG) |
15 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝑋 ∈ 𝑃) |
16 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝑌 ∈ 𝑃) |
17 | 10 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝑍 ∈ 𝑃) |
18 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝑋 ∈ (𝑍𝐼𝑌)) | |
19 | 1, 2, 3, 14, 15, 16, 17, 18 | btwncolg2 26334 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
20 | 4 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐺 ∈ TarskiG) |
21 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑋 ∈ 𝑃) |
22 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ 𝑃) |
23 | 10 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑍 ∈ 𝑃) |
24 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ (𝑋𝐼𝑍)) | |
25 | 1, 2, 3, 20, 21, 22, 23, 24 | btwncolg3 26335 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
26 | tgdim01ln.1 | . . 3 ⊢ (𝜑 → ¬ 𝐺DimTarskiG≥2) | |
27 | 1, 3, 4, 26, 6, 8, 10 | tgdim01 26285 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) |
28 | 13, 19, 25, 27 | mpjao3dan 1426 | 1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1531 ∈ wcel 2108 class class class wbr 5057 ‘cfv 6348 (class class class)co 7148 2c2 11684 Basecbs 16475 TarskiGcstrkg 26208 DimTarskiG≥cstrkgld 26212 Itvcitv 26214 LineGclng 26215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-fzo 13026 df-trkgc 26226 df-trkgcb 26228 df-trkgld 26230 df-trkg 26231 |
This theorem is referenced by: ncoltgdim2 26343 |
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