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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > llnbase | Structured version Visualization version GIF version |
Description: A lattice line is a lattice element. (Contributed by NM, 16-Jun-2012.) |
Ref | Expression |
---|---|
llnbase.b | β’ π΅ = (BaseβπΎ) |
llnbase.n | β’ π = (LLinesβπΎ) |
Ref | Expression |
---|---|
llnbase | β’ (π β π β π β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4332 | . . . 4 β’ (π β π β Β¬ π = β ) | |
2 | llnbase.n | . . . . 5 β’ π = (LLinesβπΎ) | |
3 | 2 | eqeq1i 2735 | . . . 4 β’ (π = β β (LLinesβπΎ) = β ) |
4 | 1, 3 | sylnib 327 | . . 3 β’ (π β π β Β¬ (LLinesβπΎ) = β ) |
5 | fvprc 6882 | . . 3 β’ (Β¬ πΎ β V β (LLinesβπΎ) = β ) | |
6 | 4, 5 | nsyl2 141 | . 2 β’ (π β π β πΎ β V) |
7 | llnbase.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
8 | eqid 2730 | . . . 4 β’ ( β βπΎ) = ( β βπΎ) | |
9 | eqid 2730 | . . . 4 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
10 | 7, 8, 9, 2 | islln 38680 | . . 3 β’ (πΎ β V β (π β π β (π β π΅ β§ βπ β (AtomsβπΎ)π( β βπΎ)π))) |
11 | 10 | simprbda 497 | . 2 β’ ((πΎ β V β§ π β π) β π β π΅) |
12 | 6, 11 | mpancom 684 | 1 β’ (π β π β π β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 βwrex 3068 Vcvv 3472 β c0 4321 class class class wbr 5147 βcfv 6542 Basecbs 17148 β ccvr 38435 Atomscatm 38436 LLinesclln 38665 |
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-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-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 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-llines 38672 |
This theorem is referenced by: islln2 38685 llnnleat 38687 llnneat 38688 atcvrlln2 38693 llnexatN 38695 llncmp 38696 2llnmat 38698 islpln3 38707 llnmlplnN 38713 lplnle 38714 lplnnle2at 38715 llncvrlpln2 38731 llncvrlpln 38732 2llnmj 38734 lplncmp 38736 lplnexatN 38737 lplnexllnN 38738 2llnm2N 38742 2llnm3N 38743 2llnm4 38744 2llnmeqat 38745 dalem21 38868 dalem54 38900 dalem55 38901 dalem57 38903 dalem60 38906 llnexchb2lem 39042 llnexchb2 39043 llnexch2N 39044 |
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