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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 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶) | |
2 | tgbtwnexch2.1 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) | |
3 | 2 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷)) |
4 | 1, 3 | eqeltrrd 2827 | . 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 479 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG) |
10 | tgbtwnintr.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
11 | 10 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐴 ∈ 𝑃) |
12 | tgbtwnintr.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
13 | 12 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃) |
14 | tgbtwnintr.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
15 | 14 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃) |
16 | tgbtwnintr.4 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
17 | 16 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐷 ∈ 𝑃) |
18 | simpr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶) | |
19 | tgbtwnexch2.2 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷)) | |
20 | 19 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐵𝐼𝐷)) |
21 | 2 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷)) |
22 | 5, 6, 7, 9, 15, 13, 11, 17, 20, 21 | tgbtwnintr 28420 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼𝐴)) |
23 | 5, 6, 7, 9, 15, 13, 11, 22 | tgbtwncom 28415 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶)) |
24 | 5, 6, 7, 9, 11, 13, 15, 17, 18, 23, 20 | tgbtwnouttr2 28422 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐴𝐼𝐷)) |
25 | 4, 24 | pm2.61dane 3019 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 distcds 17275 TarskiGcstrkg 28354 Itvcitv 28360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-nul 5311 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-iota 6506 df-fv 6562 df-ov 7427 df-trkgc 28375 df-trkgb 28376 df-trkgcb 28377 df-trkg 28380 |
This theorem is referenced by: tgbtwnexch 28425 tgtrisegint 28426 tgbtwnconn1lem3 28501 legtri3 28517 miriso 28597 |
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