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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 26548 | . . . 4 ⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)}) |
8 | simpl 485 | . . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑎 = 𝑢) | |
9 | 8 | breq1d 5079 | . . . . . . 7 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎𝑂𝑐 ↔ 𝑢𝑂𝑐)) |
10 | simpr 487 | . . . . . . . 8 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣) | |
11 | 10 | breq1d 5079 | . . . . . . 7 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑏𝑂𝑐 ↔ 𝑣𝑂𝑐)) |
12 | 9, 11 | anbi12d 632 | . . . . . 6 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐) ↔ (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐))) |
13 | 12 | rexbidv 3300 | . . . . 5 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐) ↔ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐))) |
14 | 13 | cbvopabv 5141 | . . . 4 ⊢ {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)} = {〈𝑢, 𝑣〉 ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)} |
15 | 7, 14 | syl6eq 2875 | . . 3 ⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {〈𝑢, 𝑣〉 ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}) |
16 | 15 | breqd 5080 | . 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 485 | . . . . . . 7 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑢 = 𝐴) | |
20 | 19 | breq1d 5079 | . . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑢𝑂𝑐 ↔ 𝐴𝑂𝑐)) |
21 | simpr 487 | . . . . . . 7 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵) | |
22 | 21 | breq1d 5079 | . . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣𝑂𝑐 ↔ 𝐵𝑂𝑐)) |
23 | 20, 22 | anbi12d 632 | . . . . 5 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐) ↔ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
24 | 23 | rexbidv 3300 | . . . 4 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐵) → (∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐) ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
25 | eqid 2824 | . . . 4 ⊢ {〈𝑢, 𝑣〉 ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)} = {〈𝑢, 𝑣〉 ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)} | |
26 | 24, 25 | brabga 5424 | . . 3 ⊢ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) → (𝐴{〈𝑢, 𝑣〉 ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
27 | 17, 18, 26 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐴{〈𝑢, 𝑣〉 ∣ ∃𝑐 ∈ 𝑃 (𝑢𝑂𝑐 ∧ 𝑣𝑂𝑐)}𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
28 | 16, 27 | bitrd 281 | 1 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 ∖ cdif 3936 class class class wbr 5069 {copab 5131 ran crn 5559 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 TarskiGcstrkg 26219 Itvcitv 26225 LineGclng 26226 hpGchpg 26546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-hpg 26547 |
This theorem is referenced by: hpgne1 26550 hpgne2 26551 lnopp2hpgb 26552 hpgid 26555 hpgcom 26556 hpgtr 26557 |
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