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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 2821 | . . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
3 | ishlg.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | hlln.1 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐺 ∈ TarskiG) |
6 | ishlg.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | 6 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐶 ∈ 𝑃) |
8 | ishlg.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | 8 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ 𝑃) |
10 | ishlg.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | 10 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐵 ∈ 𝑃) |
12 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐶𝐼𝐵)) | |
13 | 1, 2, 3, 5, 7, 9, 11, 12 | tgbtwncom 26202 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐵𝐼𝐶)) |
14 | 13 | 3mix1d 1328 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴))) |
15 | 4 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐺 ∈ TarskiG) |
16 | 6 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐶 ∈ 𝑃) |
17 | 10 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ 𝑃) |
18 | 8 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴 ∈ 𝑃) |
19 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐶𝐼𝐴)) | |
20 | 1, 2, 3, 15, 16, 17, 18, 19 | tgbtwncom 26202 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐶)) |
21 | 20 | 3mix2d 1329 | . . 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 26316 | . . . . 5 ⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))) |
25 | 22, 24 | mpbid 233 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))) |
26 | 25 | simp3d 1136 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) |
27 | 14, 21, 26 | mpjaodan 952 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴))) |
28 | hlln.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
29 | 25 | simp2d 1135 | . . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
30 | 1, 28, 3, 4, 10, 6, 29, 8 | tgellng 26267 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐿𝐶) ↔ (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴)))) |
31 | 27, 30 | mpbird 258 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 841 ∨ w3o 1078 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3016 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 Basecbs 16473 distcds 16564 TarskiGcstrkg 26144 Itvcitv 26150 LineGclng 26151 hlGchlg 26314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 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 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-trkgc 26162 df-trkgb 26163 df-trkgcb 26164 df-trkg 26167 df-hlg 26315 |
This theorem is referenced by: hlperpnel 26439 opphllem4 26464 opphl 26468 hlpasch 26470 colhp 26484 hphl 26485 trgcopy 26518 cgracgr 26532 cgraswap 26534 acopy 26547 acopyeu 26548 tgasa1 26572 |
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