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Mirrors > Home > MPE Home > Th. List > ncolne1 | Structured version Visualization version GIF version |
Description: Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) |
Ref | Expression |
---|---|
tglineelsb2.p | β’ π΅ = (BaseβπΊ) |
tglineelsb2.i | β’ πΌ = (ItvβπΊ) |
tglineelsb2.l | β’ πΏ = (LineGβπΊ) |
tglineelsb2.g | β’ (π β πΊ β TarskiG) |
ncolne.x | β’ (π β π β π΅) |
ncolne.y | β’ (π β π β π΅) |
ncolne.z | β’ (π β π β π΅) |
ncolne.2 | β’ (π β Β¬ (π β (ππΏπ) β¨ π = π)) |
Ref | Expression |
---|---|
ncolne1 | β’ (π β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ncolne.2 | . . 3 β’ (π β Β¬ (π β (ππΏπ) β¨ π = π)) | |
2 | tglineelsb2.p | . . . 4 β’ π΅ = (BaseβπΊ) | |
3 | tglineelsb2.l | . . . 4 β’ πΏ = (LineGβπΊ) | |
4 | tglineelsb2.i | . . . 4 β’ πΌ = (ItvβπΊ) | |
5 | tglineelsb2.g | . . . . 5 β’ (π β πΊ β TarskiG) | |
6 | 5 | adantr 482 | . . . 4 β’ ((π β§ π = π) β πΊ β TarskiG) |
7 | ncolne.y | . . . . 5 β’ (π β π β π΅) | |
8 | 7 | adantr 482 | . . . 4 β’ ((π β§ π = π) β π β π΅) |
9 | ncolne.z | . . . . 5 β’ (π β π β π΅) | |
10 | 9 | adantr 482 | . . . 4 β’ ((π β§ π = π) β π β π΅) |
11 | ncolne.x | . . . . 5 β’ (π β π β π΅) | |
12 | 11 | adantr 482 | . . . 4 β’ ((π β§ π = π) β π β π΅) |
13 | eqid 2733 | . . . . . 6 β’ (distβπΊ) = (distβπΊ) | |
14 | 2, 13, 4, 6, 12, 10 | tgbtwntriv1 27742 | . . . . 5 β’ ((π β§ π = π) β π β (ππΌπ)) |
15 | simpr 486 | . . . . . 6 β’ ((π β§ π = π) β π = π) | |
16 | 15 | oveq1d 7424 | . . . . 5 β’ ((π β§ π = π) β (ππΌπ) = (ππΌπ)) |
17 | 14, 16 | eleqtrd 2836 | . . . 4 β’ ((π β§ π = π) β π β (ππΌπ)) |
18 | 2, 3, 4, 6, 8, 10, 12, 17 | btwncolg1 27806 | . . 3 β’ ((π β§ π = π) β (π β (ππΏπ) β¨ π = π)) |
19 | 1, 18 | mtand 815 | . 2 β’ (π β Β¬ π = π) |
20 | 19 | neqned 2948 | 1 β’ (π β π β π) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β¨ wo 846 = wceq 1542 β wcel 2107 β wne 2941 βcfv 6544 (class class class)co 7409 Basecbs 17144 distcds 17206 TarskiGcstrkg 27678 Itvcitv 27684 LineGclng 27685 |
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-10 2138 ax-11 2155 ax-12 2172 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-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-trkgc 27699 df-trkgb 27700 df-trkgcb 27701 df-trkg 27704 |
This theorem is referenced by: ncolne2 27877 tglineneq 27895 midexlem 27943 mideulem2 27985 outpasch 28006 hlpasch 28007 trgcopy 28055 trgcopyeulem 28056 acopy 28084 acopyeu 28085 cgrg3col4 28104 tgasa1 28109 |
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