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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 2740 | . . . 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 26847 | . . 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 26961 | . 2 ⊢ (𝜑 → (𝐶(𝐾‘𝐵)𝐴 ↔ (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))))) |
15 | 1, 2, 12, 14 | mpbir3and 1341 | 1 ⊢ (𝜑 → 𝐶(𝐾‘𝐵)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 class class class wbr 5079 ‘cfv 6432 (class class class)co 7271 Basecbs 16910 distcds 16969 TarskiGcstrkg 26786 Itvcitv 26792 hlGchlg 26959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-trkgc 26807 df-trkgb 26808 df-trkgcb 26809 df-trkg 26812 df-hlg 26960 |
This theorem is referenced by: hlpasch 27115 |
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