![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2736 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
3 | ishlg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | hlln.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐺 ∈ TarskiG) |
6 | ishlg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐶 ∈ 𝑃) |
8 | hltr.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ 𝑃) |
10 | ishlg.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | 10 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐵 ∈ 𝑃) |
12 | ishlg.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
13 | 12 | adantr 481 | . . . 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 27389 | . . . . . . . 8 ⊢ (𝜑 → (𝐴(𝐾‘𝐷)𝐵 ↔ (𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ (𝐴 ∈ (𝐷𝐼𝐵) ∨ 𝐵 ∈ (𝐷𝐼𝐴))))) |
17 | 14, 16 | mpbid 231 | . . . . . . 7 ⊢ (𝜑 → (𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ (𝐴 ∈ (𝐷𝐼𝐵) ∨ 𝐵 ∈ (𝐷𝐼𝐴)))) |
18 | 17 | simp1d 1142 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐷) |
19 | 18 | necomd 2997 | . . . . 5 ⊢ (𝜑 → 𝐷 ≠ 𝐴) |
20 | 19 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ≠ 𝐴) |
21 | btwnhl.3 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐶)) | |
22 | 21 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐴𝐼𝐶)) |
23 | 1, 2, 3, 5, 13, 9, 7, 22 | tgbtwncom 27275 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐶𝐼𝐴)) |
24 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐴 ∈ (𝐷𝐼𝐵)) | |
25 | 1, 2, 3, 5, 7, 9, 13, 11, 20, 23, 24 | tgbtwnouttr 27284 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐶𝐼𝐵)) |
26 | 1, 2, 3, 5, 7, 9, 11, 25 | tgbtwncom 27275 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐵𝐼𝐶)) |
27 | 4 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐺 ∈ TarskiG) |
28 | 12 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐴 ∈ 𝑃) |
29 | 10 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐵 ∈ 𝑃) |
30 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐷 ∈ 𝑃) |
31 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐶 ∈ 𝑃) |
32 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐵 ∈ (𝐷𝐼𝐴)) | |
33 | 1, 2, 3, 27, 30, 29, 28, 32 | tgbtwncom 27275 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐷)) |
34 | 21 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐷 ∈ (𝐴𝐼𝐶)) |
35 | 1, 2, 3, 27, 28, 29, 30, 31, 33, 34 | tgbtwnexch3 27281 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐷 ∈ (𝐵𝐼𝐶)) |
36 | 17 | simp3d 1144 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐷𝐼𝐵) ∨ 𝐵 ∈ (𝐷𝐼𝐴))) |
37 | 26, 35, 36 | mpjaodan 957 | 1 ⊢ (𝜑 → 𝐷 ∈ (𝐵𝐼𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 class class class wbr 5103 ‘cfv 6493 (class class class)co 7351 Basecbs 17037 distcds 17096 TarskiGcstrkg 27214 Itvcitv 27220 hlGchlg 27387 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-trkgc 27235 df-trkgb 27236 df-trkgcb 27237 df-trkg 27240 df-hlg 27388 |
This theorem is referenced by: hlcgreulem 27404 opphllem5 27538 colhp 27557 cgrabtwn 27613 sacgr 27618 inaghl 27632 |
Copyright terms: Public domain | W3C validator |