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| Mirrors > Home > MPE Home > Th. List > lnoppnhpg | Structured version Visualization version GIF version | ||
| Description: If two points lie on the opposite side of a line 𝐷, they are not on the same half-plane. Theorem 9.9 of [Schwabhauser] p. 72. (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) | 
| lnoppnhpg.1 | ⊢ (𝜑 → 𝐴𝑂𝐵) | 
| Ref | Expression | 
|---|---|
| lnoppnhpg | ⊢ (𝜑 → ¬ 𝐴((hpG‘𝐺)‘𝐷)𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ishpg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
| 2 | eqid 2737 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
| 3 | ishpg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
| 4 | ishpg.o | . . 3 ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | |
| 5 | ishpg.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
| 6 | ishpg.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
| 7 | ishpg.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 8 | hpgbr.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 9 | 1, 2, 3, 4, 5, 6, 7, 8 | oppnid 28754 | . 2 ⊢ (𝜑 → ¬ 𝐵𝑂𝐵) | 
| 10 | hpgbr.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 11 | lnoppnhpg.1 | . . 3 ⊢ (𝜑 → 𝐴𝑂𝐵) | |
| 12 | 1, 3, 5, 4, 7, 6, 10, 8, 8, 11 | lnopp2hpgb 28771 | . 2 ⊢ (𝜑 → (𝐵𝑂𝐵 ↔ 𝐴((hpG‘𝐺)‘𝐷)𝐵)) | 
| 13 | 9, 12 | mtbid 324 | 1 ⊢ (𝜑 → ¬ 𝐴((hpG‘𝐺)‘𝐷)𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∖ cdif 3948 class class class wbr 5143 {copab 5205 ran crn 5686 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 LineGclng 28442 hpGchpg 28765 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkgld 28460 df-trkg 28461 df-cgrg 28519 df-leg 28591 df-hlg 28609 df-mir 28661 df-rag 28702 df-perpg 28704 df-hpg 28766 | 
| This theorem is referenced by: (None) | 
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