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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 789 | . . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝑃 ∧ 𝐴𝑂𝑐)) → 𝐴𝑂𝑐) | |
2 | 1, 1 | jca 507 | . . 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 26074 | . . 3 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 𝐴𝑂𝑐) |
12 | 2, 11 | reximddv 3226 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐴𝑂𝑐)) |
13 | 3, 4, 5, 9, 6, 7, 8, 8 | hpgbr 26069 | . 2 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐴 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐴𝑂𝑐))) |
14 | 12, 13 | mpbird 249 | 1 ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 ∖ cdif 3795 class class class wbr 4873 {copab 4935 ran crn 5343 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 TarskiGcstrkg 25742 Itvcitv 25748 LineGclng 25749 hpGchpg 26066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-xnn0 11691 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-hash 13411 df-word 13575 df-concat 13631 df-s1 13656 df-s2 13969 df-s3 13970 df-trkgc 25760 df-trkgb 25761 df-trkgcb 25762 df-trkg 25765 df-cgrg 25823 df-hpg 26067 |
This theorem is referenced by: tgasa1 26157 |
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