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| Mirrors > Home > MPE Home > Th. List > ncolrot1 | Structured version Visualization version GIF version | ||
| Description: Rotating 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 |
|---|---|
| ncolrot1 | 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 480 | . . 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 480 | . . 3 β’ ((π β§ (π β (ππΏπ) β¨ π = π)) β π β π) |
| 9 | tgcolg.z | . . . 4 β’ (π β π β π) | |
| 10 | 9 | adantr 480 | . . 3 β’ ((π β§ (π β (ππΏπ) β¨ π = π)) β π β π) |
| 11 | tglngval.x | . . . 4 β’ (π β π β π) | |
| 12 | 11 | adantr 480 | . . 3 β’ ((π β§ (π β (ππΏπ) β¨ π = π)) β π β π) |
| 13 | simpr 484 | . . 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 | colrot2 28235 | . 2 β’ ((π β§ (π β (ππΏπ) β¨ π = π)) β (π β (ππΏπ) β¨ π = π)) |
| 15 | 1, 14 | mtand 813 | 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 395 β¨ wo 844 = wceq 1533 β wcel 2098 βcfv 6533 (class class class)co 7401 Basecbs 17140 TarskiGcstrkg 28102 Itvcitv 28108 LineGclng 28109 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6485 df-fun 6535 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-trkgc 28123 df-trkgb 28124 df-trkgcb 28125 df-trkg 28128 |
| This theorem is referenced by: outpasch 28430 acopy 28508 cgrg3col4 28528 isoas 28539 |
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