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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 2740 | . . . . 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 26847 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐵𝐼𝐶)) |
14 | 13 | 3mix1d 1335 | . . 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 26847 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐶)) |
21 | 20 | 3mix2d 1336 | . . 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 26961 | . . . . 5 ⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))) |
25 | 22, 24 | mpbid 231 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))) |
26 | 25 | simp3d 1143 | . . 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 1142 | . . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
30 | 1, 28, 3, 4, 10, 6, 29, 8 | tgellng 26912 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐿𝐶) ↔ (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴)))) |
31 | 27, 30 | mpbird 256 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 844 ∨ w3o 1085 ∧ w3a 1086 = 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 LineGclng 26793 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-3or 1087 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-oprab 7275 df-mpo 7276 df-trkgc 26807 df-trkgb 26808 df-trkgcb 26809 df-trkg 26812 df-hlg 26960 |
This theorem is referenced by: hlperpnel 27084 opphllem4 27109 opphl 27113 hlpasch 27115 colhp 27129 hphl 27130 trgcopy 27163 cgracgr 27177 cgraswap 27179 acopy 27192 acopyeu 27193 tgasa1 27217 |
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