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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 481 | . . . 4 β’ ((π β§ π = π) β πΊ β TarskiG) |
7 | ncolne.y | . . . . 5 β’ (π β π β π΅) | |
8 | 7 | adantr 481 | . . . 4 β’ ((π β§ π = π) β π β π΅) |
9 | ncolne.z | . . . . 5 β’ (π β π β π΅) | |
10 | 9 | adantr 481 | . . . 4 β’ ((π β§ π = π) β π β π΅) |
11 | ncolne.x | . . . . 5 β’ (π β π β π΅) | |
12 | 11 | adantr 481 | . . . 4 β’ ((π β§ π = π) β π β π΅) |
13 | eqid 2736 | . . . . . 6 β’ (distβπΊ) = (distβπΊ) | |
14 | 2, 13, 4, 6, 12, 10 | tgbtwntriv1 27141 | . . . . 5 β’ ((π β§ π = π) β π β (ππΌπ)) |
15 | simpr 485 | . . . . . 6 β’ ((π β§ π = π) β π = π) | |
16 | 15 | oveq1d 7352 | . . . . 5 β’ ((π β§ π = π) β (ππΌπ) = (ππΌπ)) |
17 | 14, 16 | eleqtrd 2839 | . . . 4 β’ ((π β§ π = π) β π β (ππΌπ)) |
18 | 2, 3, 4, 6, 8, 10, 12, 17 | btwncolg1 27205 | . . 3 β’ ((π β§ π = π) β (π β (ππΏπ) β¨ π = π)) |
19 | 1, 18 | mtand 813 | . 2 β’ (π β Β¬ π = π) |
20 | 19 | neqned 2947 | 1 β’ (π β π β π) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 β¨ wo 844 = wceq 1540 β wcel 2105 β wne 2940 βcfv 6479 (class class class)co 7337 Basecbs 17009 distcds 17068 TarskiGcstrkg 27077 Itvcitv 27083 LineGclng 27084 |
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-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-sbc 3728 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-iota 6431 df-fun 6481 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-trkgc 27098 df-trkgb 27099 df-trkgcb 27100 df-trkg 27103 |
This theorem is referenced by: ncolne2 27276 tglineneq 27294 midexlem 27342 mideulem2 27384 outpasch 27405 hlpasch 27406 trgcopy 27454 trgcopyeulem 27455 acopy 27483 acopyeu 27484 cgrg3col4 27503 tgasa1 27508 |
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