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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 4298 | . . . 4 β’ (π β π β Β¬ π = β ) | |
2 | lplnbase.p | . . . . 5 β’ π = (LPlanesβπΎ) | |
3 | 2 | eqeq1i 2742 | . . . 4 β’ (π = β β (LPlanesβπΎ) = β ) |
4 | 1, 3 | sylnib 328 | . . 3 β’ (π β π β Β¬ (LPlanesβπΎ) = β ) |
5 | fvprc 6839 | . . 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 2737 | . . . 4 β’ ( β βπΎ) = ( β βπΎ) | |
9 | eqid 2737 | . . . 4 β’ (LLinesβπΎ) = (LLinesβπΎ) | |
10 | 7, 8, 9, 2 | islpln 38022 | . . 3 β’ (πΎ β V β (π β π β (π β π΅ β§ βπ₯ β (LLinesβπΎ)π₯( β βπΎ)π))) |
11 | 10 | simprbda 500 | . 2 β’ ((πΎ β V β§ π β π) β π β π΅) |
12 | 6, 11 | mpancom 687 | 1 β’ (π β π β π β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βwrex 3074 Vcvv 3448 β c0 4287 class class class wbr 5110 βcfv 6501 Basecbs 17090 β ccvr 37753 LLinesclln 37983 LPlanesclpl 37984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-lplanes 37991 |
This theorem is referenced by: islpln2 38028 llnmlplnN 38031 lplnnle2at 38033 lplnneat 38037 lplnnelln 38038 llncvrlpln2 38049 2lplnmN 38051 lplncmp 38054 lplnexatN 38055 lplnexllnN 38056 2llnjaN 38058 islvol3 38068 lvoli3 38069 lvolnle3at 38074 lplncvrlvol2 38107 lplncvrlvol 38108 lvolcmp 38109 2lplnm2N 38113 2lplnmj 38114 dalemyeb 38141 dalem10 38165 dalem16 38171 dalem44 38208 dalem55 38219 |
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