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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemcnes | Structured version Visualization version GIF version |
Description: Lemma for dath 39265. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Ref | Expression |
---|---|
dalema.ph | β’ (π β (((πΎ β HL β§ πΆ β (BaseβπΎ)) β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ (π β π β§ π β π) β§ ((Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π ) β§ Β¬ πΆ β€ (π β¨ π)) β§ (Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π)) β§ (πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π))))) |
dalemc.l | β’ β€ = (leβπΎ) |
dalemc.j | β’ β¨ = (joinβπΎ) |
dalemc.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
dalemcnes | β’ (π β πΆ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalema.ph | . . 3 β’ (π β (((πΎ β HL β§ πΆ β (BaseβπΎ)) β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ (π β π β§ π β π) β§ ((Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π ) β§ Β¬ πΆ β€ (π β¨ π)) β§ (Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π)) β§ (πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π))))) | |
2 | 1 | dalemkelat 39153 | . 2 β’ (π β πΎ β Lat) |
3 | dalemc.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | 1, 3 | dalemceb 39167 | . 2 β’ (π β πΆ β (BaseβπΎ)) |
5 | 1, 3 | dalemseb 39171 | . 2 β’ (π β π β (BaseβπΎ)) |
6 | 1, 3 | dalemteb 39172 | . 2 β’ (π β π β (BaseβπΎ)) |
7 | simp321 1320 | . . 3 β’ ((((πΎ β HL β§ πΆ β (BaseβπΎ)) β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ (π β π β§ π β π) β§ ((Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π ) β§ Β¬ πΆ β€ (π β¨ π)) β§ (Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π)) β§ (πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π)))) β Β¬ πΆ β€ (π β¨ π)) | |
8 | 1, 7 | sylbi 216 | . 2 β’ (π β Β¬ πΆ β€ (π β¨ π)) |
9 | eqid 2725 | . . 3 β’ (BaseβπΎ) = (BaseβπΎ) | |
10 | dalemc.l | . . 3 β’ β€ = (leβπΎ) | |
11 | dalemc.j | . . 3 β’ β¨ = (joinβπΎ) | |
12 | 9, 10, 11 | latnlej1l 18448 | . 2 β’ ((πΎ β Lat β§ (πΆ β (BaseβπΎ) β§ π β (BaseβπΎ) β§ π β (BaseβπΎ)) β§ Β¬ πΆ β€ (π β¨ π)) β πΆ β π) |
13 | 2, 4, 5, 6, 8, 12 | syl131anc 1380 | 1 β’ (π β πΆ β π) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 class class class wbr 5143 βcfv 6543 (class class class)co 7416 Basecbs 17179 lecple 17239 joincjn 18302 Latclat 18422 Atomscatm 38791 HLchlt 38878 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
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 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-lub 18337 df-join 18339 df-lat 18423 df-ats 38795 df-atl 38826 df-cvlat 38850 df-hlat 38879 |
This theorem is referenced by: dalem25 39227 |
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