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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 | . . 3 ⊢ (𝜑 → 𝐵(𝐾‘𝐴)𝐶) | |
2 | hphl.1 | . . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷) | |
3 | 1, 2 | jca 507 | . 2 ⊢ (𝜑 → (𝐵(𝐾‘𝐴)𝐶 ∧ ¬ 𝐵 ∈ 𝐷)) |
4 | hpgid.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
5 | hpgid.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
6 | hpgid.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
7 | hpgid.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
8 | hpgid.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
9 | hphl.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
10 | hpgid.o | . . 3 ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | |
11 | hphl.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
12 | hphl.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
13 | hpgid.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
14 | hphl.k | . . . . . 6 ⊢ 𝐾 = (hlG‘𝐺) | |
15 | 4, 5, 14, 9, 11, 13, 7, 6, 1 | hlln 25958 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐿𝐴)) |
16 | 15 | orcd 862 | . . . 4 ⊢ (𝜑 → (𝐵 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴)) |
17 | 4, 6, 5, 7, 11, 13, 9, 16 | colrot2 25911 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) |
18 | 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 14 | colhp 26118 | . 2 ⊢ (𝜑 → (𝐵((hpG‘𝐺)‘𝐷)𝐶 ↔ (𝐵(𝐾‘𝐴)𝐶 ∧ ¬ 𝐵 ∈ 𝐷))) |
19 | 3, 18 | mpbird 249 | 1 ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 ∖ cdif 3789 class class class wbr 4886 {copab 4948 ran crn 5356 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 TarskiGcstrkg 25781 Itvcitv 25787 LineGclng 25788 hlGchlg 25951 hpGchpg 26105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-xnn0 11715 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-hash 13436 df-word 13600 df-concat 13661 df-s1 13686 df-s2 13999 df-s3 14000 df-trkgc 25799 df-trkgb 25800 df-trkgcb 25801 df-trkgld 25803 df-trkg 25804 df-cgrg 25862 df-leg 25934 df-hlg 25952 df-mir 26004 df-rag 26045 df-perpg 26047 df-hpg 26106 |
This theorem is referenced by: acopyeu 26183 tgasa1 26207 |
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