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| Mirrors > Home > MPE Home > Th. List > islnoppd | Structured version Visualization version GIF version | ||
| Description: Deduce that 𝐴 and 𝐵 lie on opposite sides of line 𝐿. (Contributed by Thierry Arnoux, 16-Aug-2020.) | 
| Ref | Expression | 
|---|---|
| hpg.p | ⊢ 𝑃 = (Base‘𝐺) | 
| hpg.d | ⊢ − = (dist‘𝐺) | 
| hpg.i | ⊢ 𝐼 = (Itv‘𝐺) | 
| hpg.o | ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | 
| islnoppd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) | 
| islnoppd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) | 
| islnoppd.c | ⊢ (𝜑 → 𝐶 ∈ 𝐷) | 
| islnoppd.1 | ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) | 
| islnoppd.2 | ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷) | 
| islnoppd.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵)) | 
| Ref | Expression | 
|---|---|
| islnoppd | ⊢ (𝜑 → 𝐴𝑂𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | islnoppd.1 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) | |
| 2 | islnoppd.2 | . . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷) | |
| 3 | islnoppd.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐷) | |
| 4 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 = 𝐶) → 𝑡 = 𝐶) | |
| 5 | 4 | eleq1d 2826 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 = 𝐶) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝐶 ∈ (𝐴𝐼𝐵))) | 
| 6 | islnoppd.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵)) | |
| 7 | 3, 5, 6 | rspcedvd 3624 | . . 3 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)) | 
| 8 | 1, 2, 7 | jca31 514 | . 2 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))) | 
| 9 | hpg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
| 10 | hpg.d | . . 3 ⊢ − = (dist‘𝐺) | |
| 11 | hpg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
| 12 | hpg.o | . . 3 ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | |
| 13 | islnoppd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 14 | islnoppd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 15 | 9, 10, 11, 12, 13, 14 | islnopp 28747 | . 2 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))) | 
| 16 | 8, 15 | mpbird 257 | 1 ⊢ (𝜑 → 𝐴𝑂𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∖ cdif 3948 class class class wbr 5143 {copab 5205 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 Itvcitv 28441 | 
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-iota 6514 df-fv 6569 df-ov 7434 | 
| This theorem is referenced by: opphllem2 28756 opphllem4 28758 outpasch 28763 lmiopp 28810 | 
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