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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 2736 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
3 | ishlg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | hlln.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐺 ∈ TarskiG) |
6 | ishlg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | 6 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐶 ∈ 𝑃) |
8 | hltr.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | 8 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ 𝑃) |
10 | ishlg.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | 10 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐵 ∈ 𝑃) |
12 | ishlg.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
13 | 12 | adantr 484 | . . . 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 26647 | . . . . . . . 8 ⊢ (𝜑 → (𝐴(𝐾‘𝐷)𝐵 ↔ (𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ (𝐴 ∈ (𝐷𝐼𝐵) ∨ 𝐵 ∈ (𝐷𝐼𝐴))))) |
17 | 14, 16 | mpbid 235 | . . . . . . 7 ⊢ (𝜑 → (𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ (𝐴 ∈ (𝐷𝐼𝐵) ∨ 𝐵 ∈ (𝐷𝐼𝐴)))) |
18 | 17 | simp1d 1144 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐷) |
19 | 18 | necomd 2987 | . . . . 5 ⊢ (𝜑 → 𝐷 ≠ 𝐴) |
20 | 19 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ≠ 𝐴) |
21 | btwnhl.3 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐶)) | |
22 | 21 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐴𝐼𝐶)) |
23 | 1, 2, 3, 5, 13, 9, 7, 22 | tgbtwncom 26533 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐶𝐼𝐴)) |
24 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐴 ∈ (𝐷𝐼𝐵)) | |
25 | 1, 2, 3, 5, 7, 9, 13, 11, 20, 23, 24 | tgbtwnouttr 26542 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐶𝐼𝐵)) |
26 | 1, 2, 3, 5, 7, 9, 11, 25 | tgbtwncom 26533 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐵𝐼𝐶)) |
27 | 4 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐺 ∈ TarskiG) |
28 | 12 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐴 ∈ 𝑃) |
29 | 10 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐵 ∈ 𝑃) |
30 | 8 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐷 ∈ 𝑃) |
31 | 6 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐶 ∈ 𝑃) |
32 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐵 ∈ (𝐷𝐼𝐴)) | |
33 | 1, 2, 3, 27, 30, 29, 28, 32 | tgbtwncom 26533 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐷)) |
34 | 21 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐷 ∈ (𝐴𝐼𝐶)) |
35 | 1, 2, 3, 27, 28, 29, 30, 31, 33, 34 | tgbtwnexch3 26539 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐷 ∈ (𝐵𝐼𝐶)) |
36 | 17 | simp3d 1146 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐷𝐼𝐵) ∨ 𝐵 ∈ (𝐷𝐼𝐴))) |
37 | 26, 35, 36 | mpjaodan 959 | 1 ⊢ (𝜑 → 𝐷 ∈ (𝐵𝐼𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 847 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 distcds 16758 TarskiGcstrkg 26475 Itvcitv 26481 hlGchlg 26645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-trkgc 26493 df-trkgb 26494 df-trkgcb 26495 df-trkg 26498 df-hlg 26646 |
This theorem is referenced by: hlcgreulem 26662 opphllem5 26796 colhp 26815 cgrabtwn 26871 sacgr 26876 inaghl 26890 |
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