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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 | β’ π = (BaseβπΊ) |
hpgid.i | β’ πΌ = (ItvβπΊ) |
hpgid.l | β’ πΏ = (LineGβπΊ) |
hpgid.g | β’ (π β πΊ β TarskiG) |
hpgid.d | β’ (π β π· β ran πΏ) |
hpgid.a | β’ (π β π΄ β π) |
hpgid.o | β’ π = {β¨π, πβ© β£ ((π β (π β π·) β§ π β (π β π·)) β§ βπ‘ β π· π‘ β (ππΌπ))} |
hpgid.1 | β’ (π β Β¬ π΄ β π·) |
Ref | Expression |
---|---|
hpgid | β’ (π β π΄((hpGβπΊ)βπ·)π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 771 | . . . 4 β’ ((π β§ (π β π β§ π΄ππ)) β π΄ππ) | |
2 | 1, 1 | jca 513 | . . 3 β’ ((π β§ (π β π β§ π΄ππ)) β (π΄ππ β§ π΄ππ)) |
3 | hpgid.p | . . . 4 β’ π = (BaseβπΊ) | |
4 | hpgid.i | . . . 4 β’ πΌ = (ItvβπΊ) | |
5 | hpgid.l | . . . 4 β’ πΏ = (LineGβπΊ) | |
6 | hpgid.g | . . . 4 β’ (π β πΊ β TarskiG) | |
7 | hpgid.d | . . . 4 β’ (π β π· β ran πΏ) | |
8 | hpgid.a | . . . 4 β’ (π β π΄ β π) | |
9 | hpgid.o | . . . 4 β’ π = {β¨π, πβ© β£ ((π β (π β π·) β§ π β (π β π·)) β§ βπ‘ β π· π‘ β (ππΌπ))} | |
10 | hpgid.1 | . . . 4 β’ (π β Β¬ π΄ β π·) | |
11 | 3, 4, 5, 6, 7, 8, 9, 10 | hpgerlem 27168 | . . 3 β’ (π β βπ β π π΄ππ) |
12 | 2, 11 | reximddv 3165 | . 2 β’ (π β βπ β π (π΄ππ β§ π΄ππ)) |
13 | 3, 4, 5, 9, 6, 7, 8, 8 | hpgbr 27163 | . 2 β’ (π β (π΄((hpGβπΊ)βπ·)π΄ β βπ β π (π΄ππ β§ π΄ππ))) |
14 | 12, 13 | mpbird 258 | 1 β’ (π β π΄((hpGβπΊ)βπ·)π΄) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 = wceq 1539 β wcel 2104 βwrex 3071 β cdif 3890 class class class wbr 5082 {copab 5144 ran crn 5598 βcfv 6455 (class class class)co 7304 Basecbs 16954 TarskiGcstrkg 26830 Itvcitv 26836 LineGclng 26837 hpGchpg 27160 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5219 ax-sep 5233 ax-nul 5240 ax-pow 5298 ax-pr 5362 ax-un 7617 ax-cnex 10970 ax-resscn 10971 ax-1cn 10972 ax-icn 10973 ax-addcl 10974 ax-addrcl 10975 ax-mulcl 10976 ax-mulrcl 10977 ax-mulcom 10978 ax-addass 10979 ax-mulass 10980 ax-distr 10981 ax-i2m1 10982 ax-1ne0 10983 ax-1rid 10984 ax-rnegex 10985 ax-rrecex 10986 ax-cnre 10987 ax-pre-lttri 10988 ax-pre-lttrn 10989 ax-pre-ltadd 10990 ax-pre-mulgt0 10991 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3279 df-rab 3280 df-v 3440 df-sbc 3723 df-csb 3839 df-dif 3896 df-un 3898 df-in 3900 df-ss 3910 df-pss 3912 df-nul 4264 df-if 4467 df-pw 4542 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4846 df-int 4888 df-iun 4934 df-br 5083 df-opab 5145 df-mpt 5166 df-tr 5200 df-id 5497 df-eprel 5503 df-po 5511 df-so 5512 df-fr 5552 df-we 5554 df-xp 5603 df-rel 5604 df-cnv 5605 df-co 5606 df-dm 5607 df-rn 5608 df-res 5609 df-ima 5610 df-pred 6214 df-ord 6281 df-on 6282 df-lim 6283 df-suc 6284 df-iota 6407 df-fun 6457 df-fn 6458 df-f 6459 df-f1 6460 df-fo 6461 df-f1o 6462 df-fv 6463 df-riota 7261 df-ov 7307 df-oprab 7308 df-mpo 7309 df-om 7742 df-1st 7860 df-2nd 7861 df-frecs 8125 df-wrecs 8156 df-recs 8230 df-rdg 8269 df-1o 8325 df-oadd 8329 df-er 8526 df-pm 8646 df-en 8762 df-dom 8763 df-sdom 8764 df-fin 8765 df-dju 9700 df-card 9738 df-pnf 11054 df-mnf 11055 df-xr 11056 df-ltxr 11057 df-le 11058 df-sub 11250 df-neg 11251 df-nn 12017 df-2 12079 df-3 12080 df-n0 12277 df-xnn0 12349 df-z 12363 df-uz 12626 df-fz 13283 df-fzo 13426 df-hash 14088 df-word 14260 df-concat 14316 df-s1 14343 df-s2 14603 df-s3 14604 df-trkgc 26851 df-trkgb 26852 df-trkgcb 26853 df-trkg 26856 df-cgrg 26914 df-hpg 27161 |
This theorem is referenced by: tgasa1 27261 |
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