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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 28474 | . . 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 2841 | . 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 28495 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐶)) |
| 17 | 1, 2, 3, 4, 13, 6, 14, 6, 14, 15, 16 | axtgpasch 28475 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) |
| 18 | 12, 17 | r19.29a 3162 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkg 28461 |
| This theorem is referenced by: tgbtwncomb 28497 tgbtwntriv1 28499 tgbtwnexch3 28502 tgbtwnexch2 28504 tgbtwnouttr 28505 tgbtwnexch 28506 tgtrisegint 28507 tgifscgr 28516 tgcgrxfr 28526 tgbtwnconn1lem1 28580 tgbtwnconn1lem2 28581 tgbtwnconn1lem3 28582 tgbtwnconn1 28583 tgbtwnconn3 28585 tgbtwnconn22 28587 tgbtwnconnln1 28588 tgbtwnconnln2 28589 legtri3 28598 legtrid 28599 legbtwn 28602 tgcgrsub2 28603 hlln 28615 btwnhl2 28621 btwnhl 28622 hlcgrex 28624 hlcgreulem 28625 tglineeltr 28639 mirreu3 28662 mirmir 28670 mireq 28673 miriso 28678 mirconn 28686 mirbtwnhl 28688 mirhl2 28689 mircgrextend 28690 miduniq 28693 colmid 28696 krippenlem 28698 krippen 28699 midexlem 28700 ragflat 28712 ragcgr 28715 footexALT 28726 footexlem1 28727 footexlem2 28728 colperpexlem1 28738 colperpexlem3 28740 mideulem2 28742 opphllem 28743 midex 28745 oppcom 28752 opphllem5 28759 opphllem6 28760 outpasch 28763 hlpasch 28764 lnopp2hpgb 28771 colhp 28778 midbtwn 28787 hypcgrlem1 28807 hypcgrlem2 28808 flatcgra 28832 cgrabtwn 28834 cgracol 28836 dfcgra2 28838 sacgr 28839 oacgr 28840 inagswap 28849 inaghl 28853 |
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