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Mirrors > Home > MPE Home > Th. List > btwnhl | Structured version Visualization version GIF version |
Description: Swap betweenness for a half-line. (Contributed by Thierry Arnoux, 2-Mar-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) |
hltr.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
btwnhl.1 | ⊢ (𝜑 → 𝐴(𝐾‘𝐷)𝐵) |
btwnhl.3 | ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐶)) |
Ref | Expression |
---|---|
btwnhl | ⊢ (𝜑 → 𝐷 ∈ (𝐵𝐼𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishlg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | eqid 2734 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
3 | ishlg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | hlln.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐺 ∈ TarskiG) |
6 | ishlg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐶 ∈ 𝑃) |
8 | hltr.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ 𝑃) |
10 | ishlg.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐵 ∈ 𝑃) |
12 | ishlg.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
13 | 12 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐴 ∈ 𝑃) |
14 | btwnhl.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐴(𝐾‘𝐷)𝐵) | |
15 | ishlg.k | . . . . . . . . 9 ⊢ 𝐾 = (hlG‘𝐺) | |
16 | 1, 3, 15, 12, 10, 8, 4 | ishlg 28624 | . . . . . . . 8 ⊢ (𝜑 → (𝐴(𝐾‘𝐷)𝐵 ↔ (𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ (𝐴 ∈ (𝐷𝐼𝐵) ∨ 𝐵 ∈ (𝐷𝐼𝐴))))) |
17 | 14, 16 | mpbid 232 | . . . . . . 7 ⊢ (𝜑 → (𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ (𝐴 ∈ (𝐷𝐼𝐵) ∨ 𝐵 ∈ (𝐷𝐼𝐴)))) |
18 | 17 | simp1d 1141 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐷) |
19 | 18 | necomd 2993 | . . . . 5 ⊢ (𝜑 → 𝐷 ≠ 𝐴) |
20 | 19 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ≠ 𝐴) |
21 | btwnhl.3 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐶)) | |
22 | 21 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐴𝐼𝐶)) |
23 | 1, 2, 3, 5, 13, 9, 7, 22 | tgbtwncom 28510 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐶𝐼𝐴)) |
24 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐴 ∈ (𝐷𝐼𝐵)) | |
25 | 1, 2, 3, 5, 7, 9, 13, 11, 20, 23, 24 | tgbtwnouttr 28519 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐶𝐼𝐵)) |
26 | 1, 2, 3, 5, 7, 9, 11, 25 | tgbtwncom 28510 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐵𝐼𝐶)) |
27 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐺 ∈ TarskiG) |
28 | 12 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐴 ∈ 𝑃) |
29 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐵 ∈ 𝑃) |
30 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐷 ∈ 𝑃) |
31 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐶 ∈ 𝑃) |
32 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐵 ∈ (𝐷𝐼𝐴)) | |
33 | 1, 2, 3, 27, 30, 29, 28, 32 | tgbtwncom 28510 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐷)) |
34 | 21 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐷 ∈ (𝐴𝐼𝐶)) |
35 | 1, 2, 3, 27, 28, 29, 30, 31, 33, 34 | tgbtwnexch3 28516 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐷 ∈ (𝐵𝐼𝐶)) |
36 | 17 | simp3d 1143 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐷𝐼𝐵) ∨ 𝐵 ∈ (𝐷𝐼𝐴))) |
37 | 26, 35, 36 | mpjaodan 960 | 1 ⊢ (𝜑 → 𝐷 ∈ (𝐵𝐼𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 distcds 17306 TarskiGcstrkg 28449 Itvcitv 28455 hlGchlg 28622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-trkgc 28470 df-trkgb 28471 df-trkgcb 28472 df-trkg 28475 df-hlg 28623 |
This theorem is referenced by: hlcgreulem 28639 opphllem5 28773 colhp 28792 cgrabtwn 28848 sacgr 28853 inaghl 28867 |
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