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Mirrors > Home > MPE Home > Th. List > hpgid | Structured version Visualization version GIF version |
Description: The half-plane relation is reflexive. Theorem 9.11 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
Ref | Expression |
---|---|
hpgid.p | ⊢ 𝑃 = (Base‘𝐺) |
hpgid.i | ⊢ 𝐼 = (Itv‘𝐺) |
hpgid.l | ⊢ 𝐿 = (LineG‘𝐺) |
hpgid.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
hpgid.d | ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
hpgid.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
hpgid.o | ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} |
hpgid.1 | ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) |
Ref | Expression |
---|---|
hpgid | ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝑃 ∧ 𝐴𝑂𝑐)) → 𝐴𝑂𝑐) | |
2 | 1, 1 | jca 512 | . . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝑃 ∧ 𝐴𝑂𝑐)) → (𝐴𝑂𝑐 ∧ 𝐴𝑂𝑐)) |
3 | hpgid.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
4 | hpgid.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | hpgid.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
6 | hpgid.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | hpgid.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
8 | hpgid.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | hpgid.o | . . . 4 ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | |
10 | hpgid.1 | . . . 4 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) | |
11 | 3, 4, 5, 6, 7, 8, 9, 10 | hpgerlem 26478 | . . 3 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 𝐴𝑂𝑐) |
12 | 2, 11 | reximddv 3272 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐴𝑂𝑐)) |
13 | 3, 4, 5, 9, 6, 7, 8, 8 | hpgbr 26473 | . 2 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐴 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐴𝑂𝑐))) |
14 | 12, 13 | mpbird 258 | 1 ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 ∖ cdif 3930 class class class wbr 5057 {copab 5119 ran crn 5549 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 TarskiGcstrkg 26143 Itvcitv 26149 LineGclng 26150 hpGchpg 26470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-s2 14198 df-s3 14199 df-trkgc 26161 df-trkgb 26162 df-trkgcb 26163 df-trkg 26166 df-cgrg 26224 df-hpg 26471 |
This theorem is referenced by: tgasa1 26571 |
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