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Mirrors > Home > MPE Home > Th. List > tgbtwntriv1 | Structured version Visualization version GIF version |
Description: Betweenness always holds for the first endpoint. Theorem 3.3 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) |
Ref | Expression |
---|---|
tgbtwntriv1 | ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tkgeom.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tkgeom.d | . 2 ⊢ − = (dist‘𝐺) | |
3 | tkgeom.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tkgeom.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tgbtwntriv2.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
6 | tgbtwntriv2.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 1, 2, 3, 4, 5, 6 | tgbtwntriv2 26848 | . 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐴)) |
8 | 1, 2, 3, 4, 5, 6, 6, 7 | tgbtwncom 26849 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 distcds 16971 TarskiGcstrkg 26788 Itvcitv 26794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-nul 5230 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-ov 7278 df-trkgc 26809 df-trkgb 26810 df-trkgcb 26811 df-trkg 26814 |
This theorem is referenced by: tgldim0itv 26865 legtri3 26951 leg0 26953 legbtwn 26955 ncolne1 26986 tglnne 26989 tglinerflx1 26994 mirinv 27027 miriso 27031 colmid 27049 krippenlem 27051 colperpex 27094 outpasch 27116 hlpasch 27117 |
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