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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 2825 | . . 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 719 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐷 ∈ ran 𝐿) |
8 | ishpg.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
9 | 8 | ad2antrr 719 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐺 ∈ TarskiG) |
10 | hpgbr.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
11 | 10 | ad2antrr 719 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐴 ∈ 𝑃) |
12 | simplr 787 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝑐 ∈ 𝑃) | |
13 | simprl 789 | . . 3 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐴𝑂𝑐) | |
14 | 1, 2, 3, 4, 5, 7, 9, 11, 12, 13 | oppne1 26050 | . 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 26069 | . . 3 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
18 | 15, 17 | mpbid 224 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) |
19 | 14, 18 | r19.29a 3288 | 1 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∃wrex 3118 ∖ cdif 3795 class class class wbr 4873 {copab 4935 ran crn 5343 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 distcds 16314 TarskiGcstrkg 25742 Itvcitv 25748 LineGclng 25749 hpGchpg 26066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-hpg 26067 |
This theorem is referenced by: colhp 26079 trgcopyeulem 26114 |
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