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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 477 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 = 𝐶) → 𝑡 = 𝐶) | |
5 | 4 | eleq1d 2852 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 = 𝐶) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝐶 ∈ (𝐴𝐼𝐵))) |
6 | islnoppd.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵)) | |
7 | 3, 5, 6 | rspcedvd 3544 | . . 3 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)) |
8 | 1, 2, 7 | jca31 507 | . 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 26242 | . 2 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))) |
16 | 8, 15 | mpbird 249 | 1 ⊢ (𝜑 → 𝐴𝑂𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∃wrex 3091 ∖ cdif 3828 class class class wbr 4934 {copab 4996 ‘cfv 6193 (class class class)co 6982 Basecbs 16345 distcds 16436 Itvcitv 25939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-sep 5064 ax-nul 5071 ax-pr 5190 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3419 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4182 df-if 4354 df-sn 4445 df-pr 4447 df-op 4451 df-uni 4718 df-br 4935 df-opab 4997 df-iota 6157 df-fv 6201 df-ov 6985 |
This theorem is referenced by: opphllem2 26251 opphllem4 26253 outpasch 26258 lmiopp 26305 |
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