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Mirrors > Home > MPE Home > Th. List > islnoppd | Structured version Visualization version GIF version |
Description: Deduce that π΄ and π΅ lie on opposite sides of line πΏ. (Contributed by Thierry Arnoux, 16-Aug-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))} |
islnoppd.a | β’ (π β π΄ β π) |
islnoppd.b | β’ (π β π΅ β π) |
islnoppd.c | β’ (π β πΆ β π·) |
islnoppd.1 | β’ (π β Β¬ π΄ β π·) |
islnoppd.2 | β’ (π β Β¬ π΅ β π·) |
islnoppd.3 | β’ (π β πΆ β (π΄πΌπ΅)) |
Ref | Expression |
---|---|
islnoppd | β’ (π β π΄ππ΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islnoppd.1 | . . 3 β’ (π β Β¬ π΄ β π·) | |
2 | islnoppd.2 | . . 3 β’ (π β Β¬ π΅ β π·) | |
3 | islnoppd.c | . . . 4 β’ (π β πΆ β π·) | |
4 | simpr 486 | . . . . 5 β’ ((π β§ π‘ = πΆ) β π‘ = πΆ) | |
5 | 4 | eleq1d 2819 | . . . 4 β’ ((π β§ π‘ = πΆ) β (π‘ β (π΄πΌπ΅) β πΆ β (π΄πΌπ΅))) |
6 | islnoppd.3 | . . . 4 β’ (π β πΆ β (π΄πΌπ΅)) | |
7 | 3, 5, 6 | rspcedvd 3615 | . . 3 β’ (π β βπ‘ β π· π‘ β (π΄πΌπ΅)) |
8 | 1, 2, 7 | jca31 516 | . 2 β’ (π β ((Β¬ π΄ β π· β§ Β¬ π΅ β π·) β§ βπ‘ β π· π‘ β (π΄πΌπ΅))) |
9 | hpg.p | . . 3 β’ π = (BaseβπΊ) | |
10 | hpg.d | . . 3 β’ β = (distβπΊ) | |
11 | hpg.i | . . 3 β’ πΌ = (ItvβπΊ) | |
12 | hpg.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))} | |
13 | islnoppd.a | . . 3 β’ (π β π΄ β π) | |
14 | islnoppd.b | . . 3 β’ (π β π΅ β π) | |
15 | 9, 10, 11, 12, 13, 14 | islnopp 27990 | . 2 β’ (π β (π΄ππ΅ β ((Β¬ π΄ β π· β§ Β¬ π΅ β π·) β§ βπ‘ β π· π‘ β (π΄πΌπ΅)))) |
16 | 8, 15 | mpbird 257 | 1 β’ (π β π΄ππ΅) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwrex 3071 β cdif 3946 class class class wbr 5149 {copab 5211 βcfv 6544 (class class class)co 7409 Basecbs 17144 distcds 17206 Itvcitv 27684 |
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-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-iota 6496 df-fv 6552 df-ov 7412 |
This theorem is referenced by: opphllem2 27999 opphllem4 28001 outpasch 28006 lmiopp 28053 |
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