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Mirrors > Home > MPE Home > Th. List > oppne2 | Structured version Visualization version GIF version |
Description: Points lying on opposite sides of a line cannot be on the line. (Contributed by Thierry Arnoux, 3-Mar-2020.) |
Ref | Expression |
---|---|
hpg.p | β’ π = (BaseβπΊ) |
hpg.d | β’ β = (distβπΊ) |
hpg.i | β’ πΌ = (ItvβπΊ) |
hpg.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))} |
opphl.l | β’ πΏ = (LineGβπΊ) |
opphl.d | β’ (π β π· β ran πΏ) |
opphl.g | β’ (π β πΊ β TarskiG) |
oppcom.a | β’ (π β π΄ β π) |
oppcom.b | β’ (π β π΅ β π) |
oppcom.o | β’ (π β π΄ππ΅) |
Ref | Expression |
---|---|
oppne2 | β’ (π β Β¬ π΅ β π·) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppcom.o | . . 3 β’ (π β π΄ππ΅) | |
2 | hpg.p | . . . 4 β’ π = (BaseβπΊ) | |
3 | hpg.d | . . . 4 β’ β = (distβπΊ) | |
4 | hpg.i | . . . 4 β’ πΌ = (ItvβπΊ) | |
5 | hpg.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))} | |
6 | oppcom.a | . . . 4 β’ (π β π΄ β π) | |
7 | oppcom.b | . . . 4 β’ (π β π΅ β π) | |
8 | 2, 3, 4, 5, 6, 7 | islnopp 28423 | . . 3 β’ (π β (π΄ππ΅ β ((Β¬ π΄ β π· β§ Β¬ π΅ β π·) β§ βπ‘ β π· π‘ β (π΄πΌπ΅)))) |
9 | 1, 8 | mpbid 231 | . 2 β’ (π β ((Β¬ π΄ β π· β§ Β¬ π΅ β π·) β§ βπ‘ β π· π‘ β (π΄πΌπ΅))) |
10 | 9 | simplrd 767 | 1 β’ (π β Β¬ π΅ β π·) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 βwrex 3069 β cdif 3945 class class class wbr 5148 {copab 5210 ran crn 5677 βcfv 6543 (class class class)co 7412 Basecbs 17151 distcds 17213 TarskiGcstrkg 28111 Itvcitv 28117 LineGclng 28118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-iota 6495 df-fv 6551 df-ov 7415 |
This theorem is referenced by: opphllem1 28431 opphllem2 28432 opphl 28438 lnopp2hpgb 28447 lnperpex 28487 |
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