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Mirrors > Home > MPE Home > Th. List > btwnhl2 | Structured version Visualization version GIF version |
Description: Deduce half-line from betweenness. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
Ref | Expression |
---|---|
ishlg.p | ⊢ 𝑃 = (Base‘𝐺) |
ishlg.i | ⊢ 𝐼 = (Itv‘𝐺) |
ishlg.k | ⊢ 𝐾 = (hlG‘𝐺) |
ishlg.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
ishlg.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
ishlg.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
hlln.1 | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
hltr.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
btwnhl1.1 | ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵)) |
btwnhl1.2 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
btwnhl2.3 | ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
Ref | Expression |
---|---|
btwnhl2 | ⊢ (𝜑 → 𝐶(𝐾‘𝐵)𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwnhl2.3 | . . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐵) | |
2 | btwnhl1.2 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
3 | ishlg.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
4 | eqid 2610 | . . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
5 | ishlg.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
6 | hlln.1 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | ishlg.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | ishlg.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
9 | ishlg.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
10 | btwnhl1.1 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵)) | |
11 | 3, 4, 5, 6, 7, 8, 9, 10 | tgbtwncom 25183 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐴)) |
12 | 11 | orcd 406 | . . 3 ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))) |
13 | 1, 2, 12 | 3jca 1235 | . 2 ⊢ (𝜑 → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))) |
14 | ishlg.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
15 | 3, 5, 14, 8, 7, 9, 6 | ishlg 25297 | . 2 ⊢ (𝜑 → (𝐶(𝐾‘𝐵)𝐴 ↔ (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))))) |
16 | 13, 15 | mpbird 246 | 1 ⊢ (𝜑 → 𝐶(𝐾‘𝐵)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 distcds 15777 TarskiGcstrkg 25129 Itvcitv 25135 hlGchlg 25295 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-trkgc 25147 df-trkgb 25148 df-trkgcb 25149 df-trkg 25152 df-hlg 25296 |
This theorem is referenced by: outpasch 25447 hlpasch 25448 |
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