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Mirrors > Home > MPE Home > Th. List > hpgbr | Structured version Visualization version GIF version |
Description: Half-planes : property for points 𝐴 and 𝐵 to belong to the same open half plane delimited by line 𝐷. Definition 9.7 of [Schwabhauser] p. 71. (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) |
Ref | Expression |
---|---|
hpgbr | ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishpg.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
2 | ishpg.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
3 | ishpg.l | . . . . 5 ⊢ 𝐿 = (LineG‘𝐺) | |
4 | ishpg.o | . . . . 5 ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | |
5 | ishpg.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | ishpg.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
7 | 1, 2, 3, 4, 5, 6 | ishpg 27101 | . . . 4 ⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)}) |
8 | simpl 482 | . . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑎 = 𝑢) | |
9 | 8 | breq1d 5088 | . . . . . . 7 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎𝑂𝑐 ↔ 𝑢𝑂𝑐)) |
10 | simpr 484 | . . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣) | |
11 | 10 | breq1d 5088 | . . . . . . 7 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑏𝑂𝑐 ↔ 𝑣𝑂𝑐)) |
12 | 9, 11 | anbi12d 630 | . . . . . 6 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐) ↔ (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐))) |
13 | 12 | rexbidv 3227 | . . . . 5 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐) ↔ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐))) |
14 | 13 | cbvopabv 5151 | . . . 4 ⊢ {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)} = {〈𝑢, 𝑣〉 ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)} |
15 | 7, 14 | eqtrdi 2795 | . . 3 ⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {〈𝑢, 𝑣〉 ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}) |
16 | 15 | breqd 5089 | . 2 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ 𝐴{〈𝑢, 𝑣〉 ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}𝐵)) |
17 | hpgbr.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
18 | hpgbr.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
19 | simpl 482 | . . . . . . 7 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑢 = 𝐴) | |
20 | 19 | breq1d 5088 | . . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑢𝑂𝑐 ↔ 𝐴𝑂𝑐)) |
21 | simpr 484 | . . . . . . 7 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵) | |
22 | 21 | breq1d 5088 | . . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣𝑂𝑐 ↔ 𝐵𝑂𝑐)) |
23 | 20, 22 | anbi12d 630 | . . . . 5 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐) ↔ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
24 | 23 | rexbidv 3227 | . . . 4 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐) ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
25 | eqid 2739 | . . . 4 ⊢ {〈𝑢, 𝑣〉 ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)} = {〈𝑢, 𝑣〉 ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)} | |
26 | 24, 25 | brabga 5448 | . . 3 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) → (𝐴{〈𝑢, 𝑣〉 ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
27 | 17, 18, 26 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝐴{〈𝑢, 𝑣〉 ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
28 | 16, 27 | bitrd 278 | 1 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ∃wrex 3066 ∖ cdif 3888 class class class wbr 5078 {copab 5140 ran crn 5589 ‘cfv 6430 (class class class)co 7268 Basecbs 16893 TarskiGcstrkg 26769 Itvcitv 26775 LineGclng 26776 hpGchpg 27099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-ov 7271 df-hpg 27100 |
This theorem is referenced by: hpgne1 27103 hpgne2 27104 lnopp2hpgb 27105 hpgid 27108 hpgcom 27109 hpgtr 27110 |
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