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Mirrors > Home > MPE Home > Th. List > tgbtwncomb | Structured version Visualization version GIF version |
Description: Betweenness commutes, biconditional version. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
Ref | Expression |
---|---|
tkgeom.p | ⊢ 𝑃 = (Base‘𝐺) |
tkgeom.d | ⊢ − = (dist‘𝐺) |
tkgeom.i | ⊢ 𝐼 = (Itv‘𝐺) |
tkgeom.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgbtwntriv2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgbtwntriv2.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tgbtwncomb.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
Ref | Expression |
---|---|
tgbtwncomb | ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ↔ 𝐵 ∈ (𝐶𝐼𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tkgeom.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tkgeom.d | . . 3 ⊢ − = (dist‘𝐺) | |
3 | tkgeom.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tkgeom.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG) |
6 | tgbtwntriv2.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐴 ∈ 𝑃) |
8 | tgbtwntriv2.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
9 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ 𝑃) |
10 | tgbtwncomb.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
11 | 10 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐶 ∈ 𝑃) |
12 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐶)) | |
13 | 1, 2, 3, 5, 7, 9, 11, 12 | tgbtwncom 27728 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐶𝐼𝐴)) |
14 | 4 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐺 ∈ TarskiG) |
15 | 10 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐶 ∈ 𝑃) |
16 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ 𝑃) |
17 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴 ∈ 𝑃) |
18 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐶𝐼𝐴)) | |
19 | 1, 2, 3, 14, 15, 16, 17, 18 | tgbtwncom 27728 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐶)) |
20 | 13, 19 | impbida 799 | 1 ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ↔ 𝐵 ∈ (𝐶𝐼𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 distcds 17202 TarskiGcstrkg 27667 Itvcitv 27673 |
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 2703 ax-nul 5305 |
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 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-ov 7408 df-trkgc 27688 df-trkgb 27689 df-trkgcb 27690 df-trkg 27693 |
This theorem is referenced by: colcom 27798 colrot1 27799 lnhl 27855 lncom 27862 lnrot1 27863 lnrot2 27864 mirreu3 27894 |
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