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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 π·, 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 | β’ π = (BaseβπΊ) |
ishpg.i | β’ πΌ = (ItvβπΊ) |
ishpg.l | β’ πΏ = (LineGβπΊ) |
ishpg.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))} |
ishpg.g | β’ (π β πΊ β TarskiG) |
ishpg.d | β’ (π β π· β ran πΏ) |
hpgbr.a | β’ (π β π΄ β π) |
hpgbr.b | β’ (π β π΅ β π) |
lnoppnhpg.1 | β’ (π β π΄ππ΅) |
Ref | Expression |
---|---|
lnoppnhpg | β’ (π β Β¬ π΄((hpGβπΊ)βπ·)π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishpg.p | . . 3 β’ π = (BaseβπΊ) | |
2 | eqid 2733 | . . 3 β’ (distβπΊ) = (distβπΊ) | |
3 | ishpg.i | . . 3 β’ πΌ = (ItvβπΊ) | |
4 | ishpg.o | . . 3 β’ π = {β¨π, πβ© β£ ((π β (π β π·) β§ π β (π β π·)) β§ βπ‘ β π· π‘ β (ππΌπ))} | |
5 | ishpg.l | . . 3 β’ πΏ = (LineGβπΊ) | |
6 | ishpg.d | . . 3 β’ (π β π· β ran πΏ) | |
7 | ishpg.g | . . 3 β’ (π β πΊ β TarskiG) | |
8 | hpgbr.b | . . 3 β’ (π β π΅ β π) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | oppnid 27730 | . 2 β’ (π β Β¬ π΅ππ΅) |
10 | hpgbr.a | . . 3 β’ (π β π΄ β π) | |
11 | lnoppnhpg.1 | . . 3 β’ (π β π΄ππ΅) | |
12 | 1, 3, 5, 4, 7, 6, 10, 8, 8, 11 | lnopp2hpgb 27747 | . 2 β’ (π β (π΅ππ΅ β π΄((hpGβπΊ)βπ·)π΅)) |
13 | 9, 12 | mtbid 324 | 1 β’ (π β Β¬ π΄((hpGβπΊ)βπ·)π΅) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwrex 3070 β cdif 3908 class class class wbr 5106 {copab 5168 ran crn 5635 βcfv 6497 (class class class)co 7358 Basecbs 17088 distcds 17147 TarskiGcstrkg 27411 Itvcitv 27417 LineGclng 27418 hpGchpg 27741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-oadd 8417 df-er 8651 df-map 8770 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-dju 9842 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-xnn0 12491 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-hash 14237 df-word 14409 df-concat 14465 df-s1 14490 df-s2 14743 df-s3 14744 df-trkgc 27432 df-trkgb 27433 df-trkgcb 27434 df-trkgld 27436 df-trkg 27437 df-cgrg 27495 df-leg 27567 df-hlg 27585 df-mir 27637 df-rag 27678 df-perpg 27680 df-hpg 27742 |
This theorem is referenced by: (None) |
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