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Mirrors > Home > MPE Home > Th. List > tgbtwnconnln1 | Structured version Visualization version GIF version |
Description: Derive colinearity from betweenness. (Contributed by Thierry Arnoux, 17-May-2019.) |
Ref | Expression |
---|---|
tgbtwnconn.p | ⊢ 𝑃 = (Base‘𝐺) |
tgbtwnconn.i | ⊢ 𝐼 = (Itv‘𝐺) |
tgbtwnconn.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgbtwnconn.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgbtwnconn.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tgbtwnconn.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
tgbtwnconn.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
tgbtwnconnln1.l | ⊢ 𝐿 = (LineG‘𝐺) |
tgbtwnconnln1.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
tgbtwnconnln1.2 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
tgbtwnconnln1.3 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) |
Ref | Expression |
---|---|
tgbtwnconnln1 | ⊢ (𝜑 → (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgbtwnconn.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tgbtwnconnln1.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
3 | tgbtwnconn.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tgbtwnconn.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐺 ∈ TarskiG) |
6 | tgbtwnconn.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | 6 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐶 ∈ 𝑃) |
8 | tgbtwnconn.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | 8 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐷 ∈ 𝑃) |
10 | tgbtwnconn.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
11 | 10 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐴 ∈ 𝑃) |
12 | simpr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐶 ∈ (𝐴𝐼𝐷)) | |
13 | 1, 2, 3, 5, 7, 9, 11, 12 | btwncolg2 25924 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷)) |
14 | 4 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG) |
15 | 6 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐶 ∈ 𝑃) |
16 | 8 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ 𝑃) |
17 | 10 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐴 ∈ 𝑃) |
18 | eqid 2778 | . . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
19 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ (𝐴𝐼𝐶)) | |
20 | 1, 18, 3, 14, 17, 16, 15, 19 | tgbtwncom 25856 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ (𝐶𝐼𝐴)) |
21 | 1, 2, 3, 14, 15, 16, 17, 20 | btwncolg3 25925 | . 2 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷)) |
22 | tgbtwnconn.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
23 | tgbtwnconnln1.1 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
24 | tgbtwnconnln1.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | |
25 | tgbtwnconnln1.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) | |
26 | 1, 3, 4, 10, 22, 6, 8, 23, 24, 25 | tgbtwnconn1 25943 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶))) |
27 | 13, 21, 26 | mpjaodan 944 | 1 ⊢ (𝜑 → (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ‘cfv 6137 (class class class)co 6924 Basecbs 16266 distcds 16358 TarskiGcstrkg 25798 Itvcitv 25804 LineGclng 25805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-pm 8145 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-xnn0 11720 df-z 11734 df-uz 11998 df-fz 12649 df-fzo 12790 df-hash 13442 df-word 13606 df-concat 13667 df-s1 13692 df-s2 14005 df-s3 14006 df-trkgc 25816 df-trkgb 25817 df-trkgcb 25818 df-trkg 25821 df-cgrg 25879 |
This theorem is referenced by: tglineeltr 25999 |
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