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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 2798 | . . . 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 26282 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐴)) |
12 | 11 | orcd 870 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))) |
13 | ishlg.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
14 | 3, 5, 13, 8, 7, 9, 6 | ishlg 26396 | . 2 ⊢ (𝜑 → (𝐶(𝐾‘𝐵)𝐴 ↔ (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))))) |
15 | 1, 2, 12, 14 | mpbir3and 1339 | 1 ⊢ (𝜑 → 𝐶(𝐾‘𝐵)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 TarskiGcstrkg 26224 Itvcitv 26230 hlGchlg 26394 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-trkgc 26242 df-trkgb 26243 df-trkgcb 26244 df-trkg 26247 df-hlg 26395 |
This theorem is referenced by: hlpasch 26550 |
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