![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2731 | . . 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 724 | . . 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 724 | . . 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 724 | . . 3 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β π΄ β π) |
12 | simplr 767 | . . 3 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β π β π) | |
13 | simprl 769 | . . 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 27780 | . 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 27799 | . . 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 3161 | 1 β’ (π β Β¬ π΄ β π·) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwrex 3069 β cdif 3925 class class class wbr 5125 {copab 5187 ran crn 5654 βcfv 6516 (class class class)co 7377 Basecbs 17109 distcds 17171 TarskiGcstrkg 27466 Itvcitv 27472 LineGclng 27473 hpGchpg 27796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-ov 7380 df-hpg 27797 |
This theorem is referenced by: colhp 27809 trgcopyeulem 27844 |
Copyright terms: Public domain | W3C validator |