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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 2737 | . . 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 28610 | . . . . . . . 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 1143 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐷) |
| 19 | 18 | necomd 2996 | . . . . 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 28496 | . . . 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 28505 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐷𝐼𝐵)) → 𝐷 ∈ (𝐶𝐼𝐵)) |
| 26 | 1, 2, 3, 5, 7, 9, 11, 25 | tgbtwncom 28496 | . 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 28496 | . . 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 28502 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐷𝐼𝐴)) → 𝐷 ∈ (𝐵𝐼𝐶)) |
| 36 | 17 | simp3d 1145 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐷𝐼𝐵) ∨ 𝐵 ∈ (𝐷𝐼𝐴))) |
| 37 | 26, 35, 36 | mpjaodan 961 | 1 ⊢ (𝜑 → 𝐷 ∈ (𝐵𝐼𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 hlGchlg 28608 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkg 28461 df-hlg 28609 |
| This theorem is referenced by: hlcgreulem 28625 opphllem5 28759 colhp 28778 cgrabtwn 28834 sacgr 28839 inaghl 28853 |
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