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| Mirrors > Home > MPE Home > Th. List > ncolcom | Structured version Visualization version GIF version | ||
| Description: Swapping non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.) |
| Ref | Expression |
|---|---|
| tglngval.p | β’ π = (BaseβπΊ) |
| tglngval.l | β’ πΏ = (LineGβπΊ) |
| tglngval.i | β’ πΌ = (ItvβπΊ) |
| tglngval.g | β’ (π β πΊ β TarskiG) |
| tglngval.x | β’ (π β π β π) |
| tglngval.y | β’ (π β π β π) |
| tgcolg.z | β’ (π β π β π) |
| ncolrot | β’ (π β Β¬ (π β (ππΏπ) β¨ π = π)) |
| Ref | Expression |
|---|---|
| ncolcom | β’ (π β Β¬ (π β (ππΏπ) β¨ π = π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ncolrot | . 2 β’ (π β Β¬ (π β (ππΏπ) β¨ π = π)) | |
| 2 | tglngval.p | . . 3 β’ π = (BaseβπΊ) | |
| 3 | tglngval.l | . . 3 β’ πΏ = (LineGβπΊ) | |
| 4 | tglngval.i | . . 3 β’ πΌ = (ItvβπΊ) | |
| 5 | tglngval.g | . . . 4 β’ (π β πΊ β TarskiG) | |
| 6 | 5 | adantr 479 | . . 3 β’ ((π β§ (π β (ππΏπ) β¨ π = π)) β πΊ β TarskiG) |
| 7 | tglngval.y | . . . 4 β’ (π β π β π) | |
| 8 | 7 | adantr 479 | . . 3 β’ ((π β§ (π β (ππΏπ) β¨ π = π)) β π β π) |
| 9 | tglngval.x | . . . 4 β’ (π β π β π) | |
| 10 | 9 | adantr 479 | . . 3 β’ ((π β§ (π β (ππΏπ) β¨ π = π)) β π β π) |
| 11 | tgcolg.z | . . . 4 β’ (π β π β π) | |
| 12 | 11 | adantr 479 | . . 3 β’ ((π β§ (π β (ππΏπ) β¨ π = π)) β π β π) |
| 13 | simpr 483 | . . 3 β’ ((π β§ (π β (ππΏπ) β¨ π = π)) β (π β (ππΏπ) β¨ π = π)) | |
| 14 | 2, 3, 4, 6, 8, 10, 12, 13 | colcom 28076 | . 2 β’ ((π β§ (π β (ππΏπ) β¨ π = π)) β (π β (ππΏπ) β¨ π = π)) |
| 15 | 1, 14 | mtand 812 | 1 β’ (π β Β¬ (π β (ππΏπ) β¨ π = π)) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β¨ wo 843 = wceq 1539 β wcel 2104 βcfv 6542 (class class class)co 7411 Basecbs 17148 TarskiGcstrkg 27945 Itvcitv 27951 LineGclng 27952 |
| 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-sep 5298 ax-nul 5305 ax-pr 5426 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 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-rab 3431 df-v 3474 df-sbc 3777 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-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-trkgc 27966 df-trkgb 27967 df-trkgcb 27968 df-trkg 27971 |
| This theorem is referenced by: ncolne2 28144 symquadlem 28207 midexlem 28210 outpasch 28273 acopyeu 28352 cgrg3col4 28371 tgasa1 28376 |
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