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Mirrors > Home > MPE Home > Th. List > opptgdim2 | Structured version Visualization version GIF version |
Description: If two points opposite to a line exist, dimension must be 2 or more. (Contributed by Thierry Arnoux, 3-Mar-2020.) |
Ref | Expression |
---|---|
hpg.p | ⊢ 𝑃 = (Base‘𝐺) |
hpg.d | ⊢ − = (dist‘𝐺) |
hpg.i | ⊢ 𝐼 = (Itv‘𝐺) |
hpg.o | ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} |
opphl.l | ⊢ 𝐿 = (LineG‘𝐺) |
opphl.d | ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
opphl.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
oppcom.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
oppcom.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
oppcom.o | ⊢ (𝜑 → 𝐴𝑂𝐵) |
Ref | Expression |
---|---|
opptgdim2 | ⊢ (𝜑 → 𝐺DimTarskiG≥2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hpg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | opphl.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
3 | hpg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | opphl.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | ad3antrrr 729 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (𝐷 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐺 ∈ TarskiG) |
6 | simpllr 775 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (𝐷 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝑃) | |
7 | simplr 768 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (𝐷 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝑃) | |
8 | oppcom.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | 8 | ad3antrrr 729 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (𝐷 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐴 ∈ 𝑃) |
10 | hpg.d | . . . . . . 7 ⊢ − = (dist‘𝐺) | |
11 | hpg.o | . . . . . . 7 ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | |
12 | opphl.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
13 | oppcom.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
14 | oppcom.o | . . . . . . 7 ⊢ (𝜑 → 𝐴𝑂𝐵) | |
15 | 1, 10, 3, 11, 2, 12, 4, 8, 13, 14 | oppne1 27982 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) |
16 | 15 | ad3antrrr 729 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (𝐷 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → ¬ 𝐴 ∈ 𝐷) |
17 | simprl 770 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (𝐷 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐷 = (𝑥𝐿𝑦)) | |
18 | 16, 17 | neleqtrd 2856 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (𝐷 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → ¬ 𝐴 ∈ (𝑥𝐿𝑦)) |
19 | simprr 772 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (𝐷 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦) | |
20 | 19 | neneqd 2946 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (𝐷 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → ¬ 𝑥 = 𝑦) |
21 | ioran 983 | . . . 4 ⊢ (¬ (𝐴 ∈ (𝑥𝐿𝑦) ∨ 𝑥 = 𝑦) ↔ (¬ 𝐴 ∈ (𝑥𝐿𝑦) ∧ ¬ 𝑥 = 𝑦)) | |
22 | 18, 20, 21 | sylanbrc 584 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (𝐷 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → ¬ (𝐴 ∈ (𝑥𝐿𝑦) ∨ 𝑥 = 𝑦)) |
23 | 1, 2, 3, 5, 6, 7, 9, 22 | ncoltgdim2 27806 | . 2 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (𝐷 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐺DimTarskiG≥2) |
24 | 1, 3, 2, 4, 12 | tgisline 27868 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝐷 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) |
25 | 23, 24 | r19.29vva 3214 | 1 ⊢ (𝜑 → 𝐺DimTarskiG≥2) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 ∖ cdif 3945 class class class wbr 5148 {copab 5210 ran crn 5677 ‘cfv 6541 (class class class)co 7406 2c2 12264 Basecbs 17141 distcds 17203 TarskiGcstrkg 27668 DimTarskiG≥cstrkgld 27672 Itvcitv 27674 LineGclng 27675 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625 df-trkgc 27689 df-trkgcb 27691 df-trkgld 27693 df-trkg 27694 |
This theorem is referenced by: opphllem5 27992 opphl 27995 |
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