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Mirrors > Home > MPE Home > Th. List > oppne1 | Structured version Visualization version GIF version |
Description: Points lying on opposite sides of a line cannot be on the line. (Contributed by Thierry Arnoux, 3-Mar-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))} |
opphl.l | ⊢ 𝐿 = (LineG‘𝐺) |
opphl.d | ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
opphl.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
oppcom.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
oppcom.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
oppcom.o | ⊢ (𝜑 → 𝐴𝑂𝐵) |
Ref | Expression |
---|---|
oppne1 | ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppcom.o | . . 3 ⊢ (𝜑 → 𝐴𝑂𝐵) | |
2 | hpg.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
3 | hpg.d | . . . 4 ⊢ − = (dist‘𝐺) | |
4 | hpg.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | hpg.o | . . . 4 ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | |
6 | oppcom.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | oppcom.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
8 | 2, 3, 4, 5, 6, 7 | islnopp 27970 | . . 3 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))) |
9 | 1, 8 | mpbid 231 | . 2 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))) |
10 | 9 | simplld 767 | 1 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 ∖ cdif 3944 class class class wbr 5147 {copab 5209 ran crn 5676 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 distcds 17202 TarskiGcstrkg 27658 Itvcitv 27664 LineGclng 27665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-iota 6492 df-fv 6548 df-ov 7407 |
This theorem is referenced by: oppne3 27974 opptgdim2 27976 opphllem1 27978 opphllem2 27979 opphllem4 27981 opphl 27985 hpgne1 27992 hpgne2 27993 lnopp2hpgb 27994 |
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