![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > 4noncolr2 | Structured version Visualization version GIF version |
Description: A way to express 4 non-colinear atoms (rotated right 2 places). (Contributed by NM, 11-Jul-2012.) |
Ref | Expression |
---|---|
3noncol.l | β’ β€ = (leβπΎ) |
3noncol.j | β’ β¨ = (joinβπΎ) |
3noncol.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
4noncolr2 | β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄) β§ (π β π β§ Β¬ π β€ (π β¨ π) β§ Β¬ π β€ ((π β¨ π) β¨ π ))) β (π β π β§ Β¬ π β€ (π β¨ π) β§ Β¬ π β€ ((π β¨ π) β¨ π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp11 1200 | . 2 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄) β§ (π β π β§ Β¬ π β€ (π β¨ π) β§ Β¬ π β€ ((π β¨ π) β¨ π ))) β πΎ β HL) | |
2 | simp13 1202 | . 2 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄) β§ (π β π β§ Β¬ π β€ (π β¨ π) β§ Β¬ π β€ ((π β¨ π) β¨ π ))) β π β π΄) | |
3 | simp2l 1196 | . 2 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄) β§ (π β π β§ Β¬ π β€ (π β¨ π) β§ Β¬ π β€ ((π β¨ π) β¨ π ))) β π β π΄) | |
4 | simp2r 1197 | . 2 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄) β§ (π β π β§ Β¬ π β€ (π β¨ π) β§ Β¬ π β€ ((π β¨ π) β¨ π ))) β π β π΄) | |
5 | simp12 1201 | . 2 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄) β§ (π β π β§ Β¬ π β€ (π β¨ π) β§ Β¬ π β€ ((π β¨ π) β¨ π ))) β π β π΄) | |
6 | 3noncol.l | . . 3 β’ β€ = (leβπΎ) | |
7 | 3noncol.j | . . 3 β’ β¨ = (joinβπΎ) | |
8 | 3noncol.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
9 | 6, 7, 8 | 4noncolr3 38954 | . 2 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄) β§ (π β π β§ Β¬ π β€ (π β¨ π) β§ Β¬ π β€ ((π β¨ π) β¨ π ))) β (π β π β§ Β¬ π β€ (π β¨ π ) β§ Β¬ π β€ ((π β¨ π ) β¨ π))) |
10 | 6, 7, 8 | 4noncolr3 38954 | . 2 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄) β§ (π β π β§ Β¬ π β€ (π β¨ π ) β§ Β¬ π β€ ((π β¨ π ) β¨ π))) β (π β π β§ Β¬ π β€ (π β¨ π) β§ Β¬ π β€ ((π β¨ π) β¨ π))) |
11 | 1, 2, 3, 4, 5, 9, 10 | syl321anc 1389 | 1 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄) β§ (π β π β§ Β¬ π β€ (π β¨ π) β§ Β¬ π β€ ((π β¨ π) β¨ π ))) β (π β π β§ Β¬ π β€ (π β¨ π) β§ Β¬ π β€ ((π β¨ π) β¨ π))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 class class class wbr 5141 βcfv 6541 (class class class)co 7414 lecple 17237 joincjn 18300 Atomscatm 38763 HLchlt 38850 |
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 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-proset 18284 df-poset 18302 df-plt 18319 df-lub 18335 df-glb 18336 df-join 18337 df-meet 18338 df-p0 18414 df-lat 18421 df-covers 38766 df-ats 38767 df-atl 38798 df-cvlat 38822 df-hlat 38851 |
This theorem is referenced by: 4noncolr1 38956 4atlem12 39113 |
Copyright terms: Public domain | W3C validator |