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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 483 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐺 ∈ TarskiG) |
6 | tgbtwnconn.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐶 ∈ 𝑃) |
8 | tgbtwnconn.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐷 ∈ 𝑃) |
10 | tgbtwnconn.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
11 | 10 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐴 ∈ 𝑃) |
12 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐶 ∈ (𝐴𝐼𝐷)) | |
13 | 1, 2, 3, 5, 7, 9, 11, 12 | btwncolg2 26342 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷)) |
14 | 4 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG) |
15 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐶 ∈ 𝑃) |
16 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ 𝑃) |
17 | 10 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐴 ∈ 𝑃) |
18 | eqid 2821 | . . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
19 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ (𝐴𝐼𝐶)) | |
20 | 1, 18, 3, 14, 17, 16, 15, 19 | tgbtwncom 26274 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ (𝐶𝐼𝐴)) |
21 | 1, 2, 3, 14, 15, 16, 17, 20 | btwncolg3 26343 | . 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 26361 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶))) |
27 | 13, 21, 26 | mpjaodan 955 | 1 ⊢ (𝜑 → (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 distcds 16574 TarskiGcstrkg 26216 Itvcitv 26222 LineGclng 26223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-s2 14210 df-s3 14211 df-trkgc 26234 df-trkgb 26235 df-trkgcb 26236 df-trkg 26239 df-cgrg 26297 |
This theorem is referenced by: tglineeltr 26417 |
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