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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 | 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))} |
hphl.k | β’ πΎ = (hlGβπΊ) |
hphl.a | β’ (π β π΄ β π·) |
hphl.b | β’ (π β π΅ β π) |
hphl.c | β’ (π β πΆ β π) |
hphl.1 | β’ (π β Β¬ π΅ β π·) |
hphl.2 | β’ (π β π΅(πΎβπ΄)πΆ) |
Ref | Expression |
---|---|
hphl | β’ (π β π΅((hpGβπΊ)βπ·)πΆ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hphl.2 | . 2 β’ (π β π΅(πΎβπ΄)πΆ) | |
2 | hphl.1 | . 2 β’ (π β Β¬ π΅ β π·) | |
3 | hpgid.p | . . 3 β’ π = (BaseβπΊ) | |
4 | hpgid.i | . . 3 β’ πΌ = (ItvβπΊ) | |
5 | hpgid.l | . . 3 β’ πΏ = (LineGβπΊ) | |
6 | hpgid.g | . . 3 β’ (π β πΊ β TarskiG) | |
7 | hpgid.d | . . 3 β’ (π β π· β ran πΏ) | |
8 | hphl.b | . . 3 β’ (π β π΅ β π) | |
9 | hpgid.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))} | |
10 | hphl.c | . . 3 β’ (π β πΆ β π) | |
11 | hphl.a | . . 3 β’ (π β π΄ β π·) | |
12 | hpgid.a | . . . 4 β’ (π β π΄ β π) | |
13 | hphl.k | . . . . . 6 β’ πΎ = (hlGβπΊ) | |
14 | 3, 4, 13, 8, 10, 12, 6, 5, 1 | hlln 28123 | . . . . 5 β’ (π β π΅ β (πΆπΏπ΄)) |
15 | 14 | orcd 869 | . . . 4 β’ (π β (π΅ β (πΆπΏπ΄) β¨ πΆ = π΄)) |
16 | 3, 5, 4, 6, 10, 12, 8, 15 | colrot2 28076 | . . 3 β’ (π β (π΄ β (π΅πΏπΆ) β¨ π΅ = πΆ)) |
17 | 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 13 | colhp 28286 | . 2 β’ (π β (π΅((hpGβπΊ)βπ·)πΆ β (π΅(πΎβπ΄)πΆ β§ Β¬ π΅ β π·))) |
18 | 1, 2, 17 | mpbir2and 709 | 1 β’ (π β π΅((hpGβπΊ)βπ·)πΆ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 βwrex 3068 β cdif 3946 class class class wbr 5149 {copab 5211 ran crn 5678 βcfv 6544 (class class class)co 7413 Basecbs 17150 TarskiGcstrkg 27943 Itvcitv 27949 LineGclng 27950 hlGchlg 28116 hpGchpg 28273 |
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 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-oadd 8474 df-er 8707 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-dju 9900 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-2 12281 df-3 12282 df-n0 12479 df-xnn0 12551 df-z 12565 df-uz 12829 df-fz 13491 df-fzo 13634 df-hash 14297 df-word 14471 df-concat 14527 df-s1 14552 df-s2 14805 df-s3 14806 df-trkgc 27964 df-trkgb 27965 df-trkgcb 27966 df-trkgld 27968 df-trkg 27969 df-cgrg 28027 df-leg 28099 df-hlg 28117 df-mir 28169 df-rag 28210 df-perpg 28212 df-hpg 28274 |
This theorem is referenced by: trgcopy 28320 acopyeu 28350 tgasa1 28374 |
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