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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 2622 | . . . . 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 477 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐶𝐼𝐵)) | |
13 | 1, 2, 3, 5, 7, 9, 11, 12 | tgbtwncom 25383 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐵𝐼𝐶)) |
14 | 13 | 3mix1d 1236 | . . 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 477 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐶𝐼𝐴)) | |
20 | 1, 2, 3, 15, 16, 17, 18, 19 | tgbtwncom 25383 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐶)) |
21 | 20 | 3mix2d 1237 | . . 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 25497 | . . . . 5 ⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))) |
25 | 22, 24 | mpbid 222 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))) |
26 | 25 | simp3d 1075 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) |
27 | 14, 21, 26 | mpjaodan 827 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴))) |
28 | hlln.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
29 | 25 | simp2d 1074 | . . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
30 | 1, 28, 3, 4, 10, 6, 29, 8 | tgellng 25448 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐿𝐶) ↔ (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴)))) |
31 | 27, 30 | mpbird 247 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 ∨ w3o 1036 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 Itvcitv 25335 LineGclng 25336 hlGchlg 25495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 df-hlg 25496 |
This theorem is referenced by: hlperpnel 25617 opphllem4 25642 opphl 25646 hlpasch 25648 colhp 25662 hphl 25663 trgcopy 25696 cgracgr 25710 cgraswap 25712 acopy 25724 acopyeu 25725 tgasa1 25739 |
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