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Mirrors > Home > MPE Home > Th. List > tgbtwnexch | Structured version Visualization version GIF version |
Description: Outer transitivity law for betweenness. Right-hand side of Theorem 3.6 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) |
tgbtwnexch.1 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
tgbtwnexch.2 | ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) |
Ref | Expression |
---|---|
tgbtwnexch | ⊢ (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 | tgbtwnexch.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) | |
10 | 1, 2, 3, 4, 7, 8, 5, 9 | tgbtwncom 27316 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐴)) |
11 | tgbtwnexch.1 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | |
12 | 1, 2, 3, 4, 7, 6, 8, 11 | tgbtwncom 27316 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴)) |
13 | 1, 2, 3, 4, 5, 8, 6, 7, 10, 12 | tgbtwnexch2 27324 | . 2 ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐴)) |
14 | 1, 2, 3, 4, 5, 6, 7, 13 | tgbtwncom 27316 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7353 Basecbs 17075 distcds 17134 TarskiGcstrkg 27255 Itvcitv 27261 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-nul 5261 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6445 df-fv 6501 df-ov 7356 df-trkgc 27276 df-trkgb 27277 df-trkgcb 27278 df-trkg 27281 |
This theorem is referenced by: tgcgrxfr 27346 tgbtwnconn1lem1 27400 tgbtwnconn1lem3 27402 legtrd 27417 hltr 27438 hlbtwn 27439 tglineeltr 27459 miriso 27498 outpasch 27583 |
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