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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lplnbase | Structured version Visualization version GIF version |
Description: A lattice plane is a lattice element. (Contributed by NM, 17-Jun-2012.) |
Ref | Expression |
---|---|
lplnbase.b | β’ π΅ = (BaseβπΎ) |
lplnbase.p | β’ π = (LPlanesβπΎ) |
Ref | Expression |
---|---|
lplnbase | 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 | lplnbase.p | . . . . 5 β’ π = (LPlanesβπΎ) | |
3 | 2 | eqeq1i 2735 | . . . 4 β’ (π = β β (LPlanesβπΎ) = β ) |
4 | 1, 3 | sylnib 327 | . . 3 β’ (π β π β Β¬ (LPlanesβπΎ) = β ) |
5 | fvprc 6882 | . . 3 β’ (Β¬ πΎ β V β (LPlanesβπΎ) = β ) | |
6 | 4, 5 | nsyl2 141 | . 2 β’ (π β π β πΎ β V) |
7 | lplnbase.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
8 | eqid 2730 | . . . 4 β’ ( β βπΎ) = ( β βπΎ) | |
9 | eqid 2730 | . . . 4 β’ (LLinesβπΎ) = (LLinesβπΎ) | |
10 | 7, 8, 9, 2 | islpln 38704 | . . 3 β’ (πΎ β V β (π β π β (π β π΅ β§ βπ₯ β (LLinesβπΎ)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 LLinesclln 38665 LPlanesclpl 38666 |
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-lplanes 38673 |
This theorem is referenced by: islpln2 38710 llnmlplnN 38713 lplnnle2at 38715 lplnneat 38719 lplnnelln 38720 llncvrlpln2 38731 2lplnmN 38733 lplncmp 38736 lplnexatN 38737 lplnexllnN 38738 2llnjaN 38740 islvol3 38750 lvoli3 38751 lvolnle3at 38756 lplncvrlvol2 38789 lplncvrlvol 38790 lvolcmp 38791 2lplnm2N 38795 2lplnmj 38796 dalemyeb 38823 dalem10 38847 dalem16 38853 dalem44 38890 dalem55 38901 |
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