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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 2821 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
3 | ishlg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | hlln.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐺 ∈ TarskiG) |
6 | ishlg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐶 ∈ 𝑃) |
8 | hltr.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ 𝑃) |
10 | ishlg.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | 10 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐵 ∈ 𝑃) |
12 | ishlg.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
13 | 12 | adantr 483 | . . . 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 26382 | . . . . . . . 8 ⊢ (𝜑 → (𝐴(𝐾‘𝐷)𝐵 ↔ (𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ (𝐴 ∈ (𝐷𝐼𝐵) ∨ 𝐵 ∈ (𝐷𝐼𝐴))))) |
17 | 14, 16 | mpbid 234 | . . . . . . 7 ⊢ (𝜑 → (𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ (𝐴 ∈ (𝐷𝐼𝐵) ∨ 𝐵 ∈ (𝐷𝐼𝐴)))) |
18 | 17 | simp1d 1138 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐷) |
19 | 18 | necomd 3071 | . . . . 5 ⊢ (𝜑 → 𝐷 ≠ 𝐴) |
20 | 19 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ≠ 𝐴) |
21 | btwnhl.3 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐶)) | |
22 | 21 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐴𝐼𝐶)) |
23 | 1, 2, 3, 5, 13, 9, 7, 22 | tgbtwncom 26268 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐶𝐼𝐴)) |
24 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐴 ∈ (𝐷𝐼𝐵)) | |
25 | 1, 2, 3, 5, 7, 9, 13, 11, 20, 23, 24 | tgbtwnouttr 26277 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐶𝐼𝐵)) |
26 | 1, 2, 3, 5, 7, 9, 11, 25 | tgbtwncom 26268 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐵𝐼𝐶)) |
27 | 4 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐺 ∈ TarskiG) |
28 | 12 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐴 ∈ 𝑃) |
29 | 10 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐵 ∈ 𝑃) |
30 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐷 ∈ 𝑃) |
31 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐶 ∈ 𝑃) |
32 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐵 ∈ (𝐷𝐼𝐴)) | |
33 | 1, 2, 3, 27, 30, 29, 28, 32 | tgbtwncom 26268 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐷)) |
34 | 21 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐷 ∈ (𝐴𝐼𝐶)) |
35 | 1, 2, 3, 27, 28, 29, 30, 31, 33, 34 | tgbtwnexch3 26274 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐷 ∈ (𝐵𝐼𝐶)) |
36 | 17 | simp3d 1140 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐷𝐼𝐵) ∨ 𝐵 ∈ (𝐷𝐼𝐴))) |
37 | 26, 35, 36 | mpjaodan 955 | 1 ⊢ (𝜑 → 𝐷 ∈ (𝐵𝐼𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 distcds 16568 TarskiGcstrkg 26210 Itvcitv 26216 hlGchlg 26380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-trkgc 26228 df-trkgb 26229 df-trkgcb 26230 df-trkg 26233 df-hlg 26381 |
This theorem is referenced by: hlcgreulem 26397 opphllem5 26531 colhp 26550 cgrabtwn 26606 sacgr 26611 inaghl 26625 |
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