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Mirrors > Home > MPE Home > Th. List > hpgne1 | Structured version Visualization version GIF version |
Description: Points on the open half plane cannot lie on its border. (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) |
hpgne1.1 | β’ (π β π΄((hpGβπΊ)βπ·)π΅) |
Ref | Expression |
---|---|
hpgne1 | β’ (π β Β¬ π΄ β π·) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishpg.p | . . 3 β’ π = (BaseβπΊ) | |
2 | eqid 2730 | . . 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 | . . . 4 β’ (π β π· β ran πΏ) | |
7 | 6 | ad2antrr 722 | . . 3 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β π· β ran πΏ) |
8 | ishpg.g | . . . 4 β’ (π β πΊ β TarskiG) | |
9 | 8 | ad2antrr 722 | . . 3 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β πΊ β TarskiG) |
10 | hpgbr.a | . . . 4 β’ (π β π΄ β π) | |
11 | 10 | ad2antrr 722 | . . 3 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β π΄ β π) |
12 | simplr 765 | . . 3 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β π β π) | |
13 | simprl 767 | . . 3 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β π΄ππ) | |
14 | 1, 2, 3, 4, 5, 7, 9, 11, 12, 13 | oppne1 28259 | . 2 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β Β¬ π΄ β π·) |
15 | hpgne1.1 | . . 3 β’ (π β π΄((hpGβπΊ)βπ·)π΅) | |
16 | hpgbr.b | . . . 4 β’ (π β π΅ β π) | |
17 | 1, 3, 5, 4, 8, 6, 10, 16 | hpgbr 28278 | . . 3 β’ (π β (π΄((hpGβπΊ)βπ·)π΅ β βπ β π (π΄ππ β§ π΅ππ))) |
18 | 15, 17 | mpbid 231 | . 2 β’ (π β βπ β π (π΄ππ β§ π΅ππ)) |
19 | 14, 18 | r19.29a 3160 | 1 β’ (π β Β¬ π΄ β π·) |
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 3944 class class class wbr 5147 {copab 5209 ran crn 5676 βcfv 6542 (class class class)co 7411 Basecbs 17148 distcds 17210 TarskiGcstrkg 27945 Itvcitv 27951 LineGclng 27952 hpGchpg 28275 |
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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 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-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-hpg 28276 |
This theorem is referenced by: colhp 28288 trgcopyeulem 28323 |
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