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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 2798 | . . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
3 | ishlg.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | hlln.1 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐺 ∈ TarskiG) |
6 | ishlg.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | 6 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐶 ∈ 𝑃) |
8 | ishlg.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | 8 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ 𝑃) |
10 | ishlg.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | 10 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐵 ∈ 𝑃) |
12 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐶𝐼𝐵)) | |
13 | 1, 2, 3, 5, 7, 9, 11, 12 | tgbtwncom 26282 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐵𝐼𝐶)) |
14 | 13 | 3mix1d 1333 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴))) |
15 | 4 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐺 ∈ TarskiG) |
16 | 6 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐶 ∈ 𝑃) |
17 | 10 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ 𝑃) |
18 | 8 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴 ∈ 𝑃) |
19 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐶𝐼𝐴)) | |
20 | 1, 2, 3, 15, 16, 17, 18, 19 | tgbtwncom 26282 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐶)) |
21 | 20 | 3mix2d 1334 | . . 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 26396 | . . . . 5 ⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))) |
25 | 22, 24 | mpbid 235 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))) |
26 | 25 | simp3d 1141 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) |
27 | 14, 21, 26 | mpjaodan 956 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴))) |
28 | hlln.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
29 | 25 | simp2d 1140 | . . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
30 | 1, 28, 3, 4, 10, 6, 29, 8 | tgellng 26347 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐿𝐶) ↔ (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴)))) |
31 | 27, 30 | mpbird 260 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 ∨ w3o 1083 ∧ w3a 1084 = 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 LineGclng 26231 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-3or 1085 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-oprab 7139 df-mpo 7140 df-trkgc 26242 df-trkgb 26243 df-trkgcb 26244 df-trkg 26247 df-hlg 26395 |
This theorem is referenced by: hlperpnel 26519 opphllem4 26544 opphl 26548 hlpasch 26550 colhp 26564 hphl 26565 trgcopy 26598 cgracgr 26612 cgraswap 26614 acopy 26627 acopyeu 26628 tgasa1 26652 |
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