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Mirrors > Home > MPE Home > Th. List > hpgne1 | Structured version Visualization version GIF version |
Description: Points on the open half plane cannot lie on its border. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
Ref | Expression |
---|---|
ishpg.p | ⊢ 𝑃 = (Base‘𝐺) |
ishpg.i | ⊢ 𝐼 = (Itv‘𝐺) |
ishpg.l | ⊢ 𝐿 = (LineG‘𝐺) |
ishpg.o | ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} |
ishpg.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
ishpg.d | ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
hpgbr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
hpgbr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
hpgne1.1 | ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵) |
Ref | Expression |
---|---|
hpgne1 | ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishpg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | eqid 2732 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
3 | ishpg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | ishpg.o | . . 3 ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | |
5 | ishpg.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
6 | ishpg.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
7 | 6 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐷 ∈ ran 𝐿) |
8 | ishpg.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
9 | 8 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐺 ∈ TarskiG) |
10 | hpgbr.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
11 | 10 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐴 ∈ 𝑃) |
12 | simplr 767 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝑐 ∈ 𝑃) | |
13 | simprl 769 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐴𝑂𝑐) | |
14 | 1, 2, 3, 4, 5, 7, 9, 11, 12, 13 | oppne1 27981 | . 2 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → ¬ 𝐴 ∈ 𝐷) |
15 | hpgne1.1 | . . 3 ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵) | |
16 | hpgbr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
17 | 1, 3, 5, 4, 8, 6, 10, 16 | hpgbr 28000 | . . 3 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
18 | 15, 17 | mpbid 231 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) |
19 | 14, 18 | r19.29a 3162 | 1 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ∖ cdif 3944 class class class wbr 5147 {copab 5209 ran crn 5676 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 distcds 17202 TarskiGcstrkg 27667 Itvcitv 27673 LineGclng 27674 hpGchpg 27997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-hpg 27998 |
This theorem is referenced by: colhp 28010 trgcopyeulem 28045 |
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