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Mirrors > Home > MPE Home > Th. List > tgbtwnexch3 | Structured version Visualization version GIF version |
Description: Exchange the first endpoint in betweenness. Left-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-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) |
tgbtwnexch3.5 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
tgbtwnexch3.6 | ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) |
Ref | Expression |
---|---|
tgbtwnexch3 | ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷)) |
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.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
6 | tgbtwnintr.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | tgbtwnintr.4 | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
8 | tgbtwnintr.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | tgbtwnexch3.5 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | |
10 | 1, 2, 3, 4, 8, 5, 6, 9 | tgbtwncom 26551 | . 2 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴)) |
11 | tgbtwnexch3.6 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) | |
12 | 1, 2, 3, 4, 8, 6, 7, 11 | tgbtwncom 26551 | . 2 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐴)) |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 12 | tgbtwnintr 26556 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 distcds 16776 TarskiGcstrkg 26493 Itvcitv 26499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-nul 5188 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-iota 6327 df-fv 6377 df-ov 7205 df-trkgc 26511 df-trkgb 26512 df-trkgcb 26513 df-trkg 26516 |
This theorem is referenced by: tgbtwnouttr2 26558 tgifscgr 26571 tgcgrxfr 26581 tgbtwnconn1lem1 26635 tgbtwnconn1lem2 26636 tgbtwnconn1lem3 26637 tgbtwnconn2 26639 tgbtwnconn3 26640 btwnhl 26677 tglineeltr 26694 miriso 26733 krippenlem 26753 outpasch 26818 hlpasch 26819 |
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