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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 725 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐺 ∈ TarskiG) |
6 | tgbtwntriv2.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | 6 | ad2antrr 725 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵 ∈ 𝑃) |
8 | simplr 768 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ 𝑃) | |
9 | simprl 770 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ (𝐵𝐼𝐵)) | |
10 | 1, 2, 3, 5, 7, 8, 9 | axtgbtwnid 26260 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵 = 𝑥) |
11 | simprr 772 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ (𝐶𝐼𝐴)) | |
12 | 10, 11 | eqeltrd 2890 | . 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 26281 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐶)) |
17 | 1, 2, 3, 4, 13, 6, 14, 6, 14, 15, 16 | axtgpasch 26261 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) |
18 | 12, 17 | r19.29a 3248 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 TarskiGcstrkg 26224 Itvcitv 26230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-trkgc 26242 df-trkgb 26243 df-trkgcb 26244 df-trkg 26247 |
This theorem is referenced by: tgbtwncomb 26283 tgbtwntriv1 26285 tgbtwnexch3 26288 tgbtwnexch2 26290 tgbtwnouttr 26291 tgbtwnexch 26292 tgtrisegint 26293 tgifscgr 26302 tgcgrxfr 26312 tgbtwnconn1lem1 26366 tgbtwnconn1lem2 26367 tgbtwnconn1lem3 26368 tgbtwnconn1 26369 tgbtwnconn3 26371 tgbtwnconn22 26373 tgbtwnconnln1 26374 tgbtwnconnln2 26375 legtri3 26384 legtrid 26385 legbtwn 26388 tgcgrsub2 26389 hlln 26401 btwnhl2 26407 btwnhl 26408 hlcgrex 26410 hlcgreulem 26411 tglineeltr 26425 mirreu3 26448 mirmir 26456 mireq 26459 miriso 26464 mirconn 26472 mirbtwnhl 26474 mirhl2 26475 mircgrextend 26476 miduniq 26479 colmid 26482 krippenlem 26484 krippen 26485 midexlem 26486 ragflat 26498 ragcgr 26501 footexALT 26512 footexlem1 26513 footexlem2 26514 colperpexlem1 26524 colperpexlem3 26526 mideulem2 26528 opphllem 26529 midex 26531 oppcom 26538 opphllem5 26545 opphllem6 26546 outpasch 26549 hlpasch 26550 lnopp2hpgb 26557 colhp 26564 midbtwn 26573 hypcgrlem1 26593 hypcgrlem2 26594 flatcgra 26618 cgrabtwn 26620 cgracol 26622 dfcgra2 26624 sacgr 26625 oacgr 26626 inagswap 26635 inaghl 26639 |
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