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Mirrors > Home > MPE Home > Th. List > hphl | Structured version Visualization version GIF version |
Description: If two points are on the same half-line with endpoint on a line, they are on the same half-plane defined by this line. (Contributed by Thierry Arnoux, 9-Aug-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))} |
hphl.k | ⊢ 𝐾 = (hlG‘𝐺) |
hphl.a | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
hphl.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
hphl.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
hphl.1 | ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷) |
hphl.2 | ⊢ (𝜑 → 𝐵(𝐾‘𝐴)𝐶) |
Ref | Expression |
---|---|
hphl | ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hphl.2 | . 2 ⊢ (𝜑 → 𝐵(𝐾‘𝐴)𝐶) | |
2 | hphl.1 | . 2 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷) | |
3 | hpgid.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
4 | hpgid.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | hpgid.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
6 | hpgid.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | hpgid.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
8 | hphl.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
9 | hpgid.o | . . 3 ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | |
10 | hphl.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
11 | hphl.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
12 | hpgid.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
13 | hphl.k | . . . . . 6 ⊢ 𝐾 = (hlG‘𝐺) | |
14 | 3, 4, 13, 8, 10, 12, 6, 5, 1 | hlln 26392 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐿𝐴)) |
15 | 14 | orcd 869 | . . . 4 ⊢ (𝜑 → (𝐵 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴)) |
16 | 3, 5, 4, 6, 10, 12, 8, 15 | colrot2 26345 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) |
17 | 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 13 | colhp 26555 | . 2 ⊢ (𝜑 → (𝐵((hpG‘𝐺)‘𝐷)𝐶 ↔ (𝐵(𝐾‘𝐴)𝐶 ∧ ¬ 𝐵 ∈ 𝐷))) |
18 | 1, 2, 17 | mpbir2and 711 | 1 ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ∖ cdif 3932 class class class wbr 5065 {copab 5127 ran crn 5555 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 TarskiGcstrkg 26215 Itvcitv 26221 LineGclng 26222 hlGchlg 26385 hpGchpg 26542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-xnn0 11967 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-concat 13922 df-s1 13949 df-s2 14209 df-s3 14210 df-trkgc 26233 df-trkgb 26234 df-trkgcb 26235 df-trkgld 26237 df-trkg 26238 df-cgrg 26296 df-leg 26368 df-hlg 26386 df-mir 26438 df-rag 26479 df-perpg 26481 df-hpg 26543 |
This theorem is referenced by: trgcopy 26589 acopyeu 26619 tgasa1 26643 |
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