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Mirrors > Home > MPE Home > Th. List > tgbtwnouttr | Structured version Visualization version GIF version |
Description: Outer transitivity law for betweenness. Right-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
Ref | Expression |
---|---|
tkgeom.p | ⊢ 𝑃 = (Base‘𝐺) |
tkgeom.d | ⊢ − = (dist‘𝐺) |
tkgeom.i | ⊢ 𝐼 = (Itv‘𝐺) |
tkgeom.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgbtwnintr.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgbtwnintr.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tgbtwnintr.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
tgbtwnintr.4 | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
tgbtwnouttr.1 | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
tgbtwnouttr.2 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
tgbtwnouttr.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷)) |
Ref | Expression |
---|---|
tgbtwnouttr | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) |
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 | tgbtwnintr.4 | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
6 | tgbtwnintr.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | tgbtwnintr.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | tgbtwnintr.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
9 | tgbtwnouttr.1 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
10 | 9 | necomd 3055 | . . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
11 | tgbtwnouttr.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷)) | |
12 | 1, 2, 3, 4, 6, 8, 5, 11 | tgbtwncom 25801 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐵)) |
13 | tgbtwnouttr.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | |
14 | 1, 2, 3, 4, 7, 6, 8, 13 | tgbtwncom 25801 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴)) |
15 | 1, 2, 3, 4, 5, 8, 6, 7, 10, 12, 14 | tgbtwnouttr2 25808 | . 2 ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐴)) |
16 | 1, 2, 3, 4, 5, 6, 7, 15 | tgbtwncom 25801 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 ‘cfv 6124 (class class class)co 6906 Basecbs 16223 distcds 16315 TarskiGcstrkg 25743 Itvcitv 25749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-nul 5014 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-iota 6087 df-fv 6132 df-ov 6909 df-trkgc 25761 df-trkgb 25762 df-trkgcb 25763 df-trkg 25766 |
This theorem is referenced by: btwnhl 25927 tglineeltr 25944 outpasch 26065 |
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