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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 724 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐺 ∈ TarskiG) |
6 | tgbtwntriv2.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | 6 | ad2antrr 724 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵 ∈ 𝑃) |
8 | simplr 767 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ 𝑃) | |
9 | simprl 769 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ (𝐵𝐼𝐵)) | |
10 | 1, 2, 3, 5, 7, 8, 9 | axtgbtwnid 26246 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵 = 𝑥) |
11 | simprr 771 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ (𝐶𝐼𝐴)) | |
12 | 10, 11 | eqeltrd 2913 | . 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 26267 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐶)) |
17 | 1, 2, 3, 4, 13, 6, 14, 6, 14, 15, 16 | axtgpasch 26247 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) |
18 | 12, 17 | r19.29a 3289 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 distcds 16568 TarskiGcstrkg 26210 Itvcitv 26216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-nul 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-iota 6309 df-fv 6358 df-ov 7153 df-trkgc 26228 df-trkgb 26229 df-trkgcb 26230 df-trkg 26233 |
This theorem is referenced by: tgbtwncomb 26269 tgbtwntriv1 26271 tgbtwnexch3 26274 tgbtwnexch2 26276 tgbtwnouttr 26277 tgbtwnexch 26278 tgtrisegint 26279 tgifscgr 26288 tgcgrxfr 26298 tgbtwnconn1lem1 26352 tgbtwnconn1lem2 26353 tgbtwnconn1lem3 26354 tgbtwnconn1 26355 tgbtwnconn3 26357 tgbtwnconn22 26359 tgbtwnconnln1 26360 tgbtwnconnln2 26361 legtri3 26370 legtrid 26371 legbtwn 26374 tgcgrsub2 26375 hlln 26387 btwnhl2 26393 btwnhl 26394 hlcgrex 26396 hlcgreulem 26397 tglineeltr 26411 mirreu3 26434 mirmir 26442 mireq 26445 miriso 26450 mirconn 26458 mirbtwnhl 26460 mirhl2 26461 mircgrextend 26462 miduniq 26465 colmid 26468 krippenlem 26470 krippen 26471 midexlem 26472 ragflat 26484 ragcgr 26487 footexALT 26498 footexlem1 26499 footexlem2 26500 colperpexlem1 26510 colperpexlem3 26512 mideulem2 26514 opphllem 26515 midex 26517 oppcom 26524 opphllem5 26531 opphllem6 26532 outpasch 26535 hlpasch 26536 lnopp2hpgb 26543 colhp 26550 midbtwn 26559 hypcgrlem1 26579 hypcgrlem2 26580 flatcgra 26604 cgrabtwn 26606 cgracol 26608 dfcgra2 26610 sacgr 26611 oacgr 26612 inagswap 26621 inaghl 26625 |
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