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Mirrors > Home > MPE Home > Th. List > hlln | Structured version Visualization version GIF version |
Description: The half-line relation implies colinearity, part of Theorem 6.4 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 22-Feb-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) |
hlln.l | ⊢ 𝐿 = (LineG‘𝐺) |
hlln.2 | ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) |
Ref | Expression |
---|---|
hlln | ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishlg.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
2 | eqid 2777 | . . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
3 | ishlg.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | hlln.1 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐺 ∈ TarskiG) |
6 | ishlg.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | 6 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐶 ∈ 𝑃) |
8 | ishlg.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | 8 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ 𝑃) |
10 | ishlg.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | 10 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐵 ∈ 𝑃) |
12 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐶𝐼𝐵)) | |
13 | 1, 2, 3, 5, 7, 9, 11, 12 | tgbtwncom 25839 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐵𝐼𝐶)) |
14 | 13 | 3mix1d 1392 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴))) |
15 | 4 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐺 ∈ TarskiG) |
16 | 6 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐶 ∈ 𝑃) |
17 | 10 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ 𝑃) |
18 | 8 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴 ∈ 𝑃) |
19 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐶𝐼𝐴)) | |
20 | 1, 2, 3, 15, 16, 17, 18, 19 | tgbtwncom 25839 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐶)) |
21 | 20 | 3mix2d 1393 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴))) |
22 | hlln.2 | . . . . 5 ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) | |
23 | ishlg.k | . . . . . 6 ⊢ 𝐾 = (hlG‘𝐺) | |
24 | 1, 3, 23, 8, 10, 6, 4 | ishlg 25953 | . . . . 5 ⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))) |
25 | 22, 24 | mpbid 224 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))) |
26 | 25 | simp3d 1135 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) |
27 | 14, 21, 26 | mpjaodan 944 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴))) |
28 | hlln.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
29 | 25 | simp2d 1134 | . . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
30 | 1, 28, 3, 4, 10, 6, 29, 8 | tgellng 25904 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐿𝐶) ↔ (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴)))) |
31 | 27, 30 | mpbird 249 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 836 ∨ w3o 1070 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 distcds 16347 TarskiGcstrkg 25781 Itvcitv 25787 LineGclng 25788 hlGchlg 25951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-trkgc 25799 df-trkgb 25800 df-trkgcb 25801 df-trkg 25804 df-hlg 25952 |
This theorem is referenced by: hlperpnel 26073 opphllem4 26098 opphl 26102 hlpasch 26104 colhp 26118 hphl 26119 trgcopy 26152 cgracgr 26166 cgraswap 26168 acopy 26182 acopyeu 26183 tgasa1 26207 |
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