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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 2798 | . . 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 725 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐷 ∈ ran 𝐿) |
8 | ishpg.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
9 | 8 | ad2antrr 725 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐺 ∈ TarskiG) |
10 | hpgbr.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
11 | 10 | ad2antrr 725 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐴 ∈ 𝑃) |
12 | simplr 768 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝑐 ∈ 𝑃) | |
13 | simprl 770 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐴𝑂𝑐) | |
14 | 1, 2, 3, 4, 5, 7, 9, 11, 12, 13 | oppne1 26535 | . 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 26554 | . . 3 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
18 | 15, 17 | mpbid 235 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) |
19 | 14, 18 | r19.29a 3248 | 1 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ∖ cdif 3878 class class class wbr 5030 {copab 5092 ran crn 5520 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 TarskiGcstrkg 26224 Itvcitv 26230 LineGclng 26231 hpGchpg 26551 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-hpg 26552 |
This theorem is referenced by: colhp 26564 trgcopyeulem 26599 |
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