Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > btwncolg3 | Structured version Visualization version GIF version |
Description: Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-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) |
btwncolg3.z | ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍)) |
Ref | Expression |
---|---|
btwncolg3 | ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwncolg3.z | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍)) | |
2 | 1 | 3mix3d 1334 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) |
3 | tglngval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
4 | tglngval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
5 | tglngval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
6 | tglngval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | tglngval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
8 | tglngval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
9 | tgcolg.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
10 | 3, 4, 5, 6, 7, 8, 9 | tgcolg 26340 | . 2 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
11 | 2, 10 | mpbird 259 | 1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 ∨ w3o 1082 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 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-sep 5203 ax-nul 5210 ax-pr 5330 |
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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-trkgc 26234 df-trkgcb 26236 df-trkg 26239 |
This theorem is referenced by: tgdim01ln 26350 lnxfr 26352 tgidinside 26357 tgbtwnconn1lem3 26360 tgbtwnconnln3 26364 tgbtwnconnln1 26366 tgbtwnconnln2 26367 legov 26371 legov2 26372 legtrd 26375 tglineeltr 26417 krippenlem 26476 midexlem 26478 footexALT 26504 footexlem2 26506 mideulem2 26520 hlpasch 26542 hypcgrlem1 26585 cgracol 26614 |
Copyright terms: Public domain | W3C validator |