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Mirrors > Home > MPE Home > Th. List > tgbtwnexch2 | Structured version Visualization version GIF version |
Description: Exchange the outer point of two betweenness statements. Right-hand side of Theorem 3.5 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) |
tgbtwnexch2.1 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) |
tgbtwnexch2.2 | ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷)) |
Ref | Expression |
---|---|
tgbtwnexch2 | ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 476 | . . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶) | |
2 | tgbtwnexch2.1 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) | |
3 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷)) |
4 | 1, 3 | eqeltrrd 2689 | . 2 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐶 ∈ (𝐴𝐼𝐷)) |
5 | tkgeom.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
6 | tkgeom.d | . . 3 ⊢ − = (dist‘𝐺) | |
7 | tkgeom.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
8 | tkgeom.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG) |
10 | tgbtwnintr.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
11 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐴 ∈ 𝑃) |
12 | tgbtwnintr.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
13 | 12 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃) |
14 | tgbtwnintr.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
15 | 14 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃) |
16 | tgbtwnintr.4 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
17 | 16 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐷 ∈ 𝑃) |
18 | simpr 476 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶) | |
19 | tgbtwnexch2.2 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷)) | |
20 | 19 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐵𝐼𝐷)) |
21 | 2 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷)) |
22 | 5, 6, 7, 9, 15, 13, 11, 17, 20, 21 | tgbtwnintr 25188 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼𝐴)) |
23 | 5, 6, 7, 9, 15, 13, 11, 22 | tgbtwncom 25183 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶)) |
24 | 5, 6, 7, 9, 11, 13, 15, 17, 18, 23, 20 | tgbtwnouttr2 25190 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐴𝐼𝐷)) |
25 | 4, 24 | pm2.61dane 2869 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 distcds 15777 TarskiGcstrkg 25129 Itvcitv 25135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-trkgc 25147 df-trkgb 25148 df-trkgcb 25149 df-trkg 25152 |
This theorem is referenced by: tgbtwnexch 25193 tgtrisegint 25194 tgbtwnconn1lem3 25269 legtri3 25285 miriso 25365 |
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