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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 | . 2 ⊢ (𝜑 → 𝐶 ≠ 𝐵) | |
2 | btwnhl1.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
3 | ishlg.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
4 | eqid 2821 | . . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
5 | ishlg.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
6 | hlln.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | ishlg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | ishlg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
9 | ishlg.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
10 | btwnhl1.1 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵)) | |
11 | 3, 4, 5, 6, 7, 8, 9, 10 | tgbtwncom 26273 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐴)) |
12 | 11 | orcd 869 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))) |
13 | ishlg.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
14 | 3, 5, 13, 8, 7, 9, 6 | ishlg 26387 | . 2 ⊢ (𝜑 → (𝐶(𝐾‘𝐵)𝐴 ↔ (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))))) |
15 | 1, 2, 12, 14 | mpbir3and 1338 | 1 ⊢ (𝜑 → 𝐶(𝐾‘𝐵)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 distcds 16573 TarskiGcstrkg 26215 Itvcitv 26221 hlGchlg 26385 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-trkgc 26233 df-trkgb 26234 df-trkgcb 26235 df-trkg 26238 df-hlg 26386 |
This theorem is referenced by: hlpasch 26541 |
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