![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 484 | . . 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 2840 | . 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 484 | . . 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 28516 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼𝐴)) |
23 | 5, 6, 7, 9, 15, 13, 11, 22 | tgbtwncom 28511 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶)) |
24 | 5, 6, 7, 9, 11, 13, 15, 17, 18, 23, 20 | tgbtwnouttr2 28518 | . 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐴𝐼𝐷)) |
25 | 4, 24 | pm2.61dane 3027 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 distcds 17307 TarskiGcstrkg 28450 Itvcitv 28456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-trkgc 28471 df-trkgb 28472 df-trkgcb 28473 df-trkg 28476 |
This theorem is referenced by: tgbtwnexch 28521 tgtrisegint 28522 tgbtwnconn1lem3 28597 legtri3 28613 miriso 28693 |
Copyright terms: Public domain | W3C validator |