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Mirrors > Home > MPE Home > Th. List > tgbtwncom | Structured version Visualization version GIF version |
Description: Betweenness commutes. Theorem 3.2 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
Ref | Expression |
---|---|
tkgeom.p | ⊢ 𝑃 = (Base‘𝐺) |
tkgeom.d | ⊢ − = (dist‘𝐺) |
tkgeom.i | ⊢ 𝐼 = (Itv‘𝐺) |
tkgeom.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgbtwntriv2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgbtwntriv2.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tgbtwncom.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
tgbtwncom.4 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
Ref | Expression |
---|---|
tgbtwncom | ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tkgeom.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tkgeom.d | . . . 4 ⊢ − = (dist‘𝐺) | |
3 | tkgeom.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tkgeom.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | ad2antrr 726 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐺 ∈ TarskiG) |
6 | tgbtwntriv2.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | 6 | ad2antrr 726 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵 ∈ 𝑃) |
8 | simplr 769 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ 𝑃) | |
9 | simprl 771 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ (𝐵𝐼𝐵)) | |
10 | 1, 2, 3, 5, 7, 8, 9 | axtgbtwnid 28488 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵 = 𝑥) |
11 | simprr 773 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ (𝐶𝐼𝐴)) | |
12 | 10, 11 | eqeltrd 2838 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵 ∈ (𝐶𝐼𝐴)) |
13 | tgbtwntriv2.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
14 | tgbtwncom.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
15 | tgbtwncom.4 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | |
16 | 1, 2, 3, 4, 6, 14 | tgbtwntriv2 28509 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐶)) |
17 | 1, 2, 3, 4, 13, 6, 14, 6, 14, 15, 16 | axtgpasch 28489 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) |
18 | 12, 17 | r19.29a 3159 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 distcds 17306 TarskiGcstrkg 28449 Itvcitv 28455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-nul 5311 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 df-ov 7433 df-trkgc 28470 df-trkgb 28471 df-trkgcb 28472 df-trkg 28475 |
This theorem is referenced by: tgbtwncomb 28511 tgbtwntriv1 28513 tgbtwnexch3 28516 tgbtwnexch2 28518 tgbtwnouttr 28519 tgbtwnexch 28520 tgtrisegint 28521 tgifscgr 28530 tgcgrxfr 28540 tgbtwnconn1lem1 28594 tgbtwnconn1lem2 28595 tgbtwnconn1lem3 28596 tgbtwnconn1 28597 tgbtwnconn3 28599 tgbtwnconn22 28601 tgbtwnconnln1 28602 tgbtwnconnln2 28603 legtri3 28612 legtrid 28613 legbtwn 28616 tgcgrsub2 28617 hlln 28629 btwnhl2 28635 btwnhl 28636 hlcgrex 28638 hlcgreulem 28639 tglineeltr 28653 mirreu3 28676 mirmir 28684 mireq 28687 miriso 28692 mirconn 28700 mirbtwnhl 28702 mirhl2 28703 mircgrextend 28704 miduniq 28707 colmid 28710 krippenlem 28712 krippen 28713 midexlem 28714 ragflat 28726 ragcgr 28729 footexALT 28740 footexlem1 28741 footexlem2 28742 colperpexlem1 28752 colperpexlem3 28754 mideulem2 28756 opphllem 28757 midex 28759 oppcom 28766 opphllem5 28773 opphllem6 28774 outpasch 28777 hlpasch 28778 lnopp2hpgb 28785 colhp 28792 midbtwn 28801 hypcgrlem1 28821 hypcgrlem2 28822 flatcgra 28846 cgrabtwn 28848 cgracol 28850 dfcgra2 28852 sacgr 28853 oacgr 28854 inagswap 28863 inaghl 28867 |
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