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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 4328 | . . . 4 β’ (π β π β Β¬ π = β ) | |
2 | lplnbase.p | . . . . 5 β’ π = (LPlanesβπΎ) | |
3 | 2 | eqeq1i 2731 | . . . 4 β’ (π = β β (LPlanesβπΎ) = β ) |
4 | 1, 3 | sylnib 328 | . . 3 β’ (π β π β Β¬ (LPlanesβπΎ) = β ) |
5 | fvprc 6877 | . . 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 2726 | . . . 4 β’ ( β βπΎ) = ( β βπΎ) | |
9 | eqid 2726 | . . . 4 β’ (LLinesβπΎ) = (LLinesβπΎ) | |
10 | 7, 8, 9, 2 | islpln 38914 | . . 3 β’ (πΎ β V β (π β π β (π β π΅ β§ βπ₯ β (LLinesβπΎ)π₯( β βπΎ)π))) |
11 | 10 | simprbda 498 | . 2 β’ ((πΎ β V β§ π β π) β π β π΅) |
12 | 6, 11 | mpancom 685 | 1 β’ (π β π β π β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βwrex 3064 Vcvv 3468 β c0 4317 class class class wbr 5141 βcfv 6537 Basecbs 17153 β ccvr 38645 LLinesclln 38875 LPlanesclpl 38876 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 df-lplanes 38883 |
This theorem is referenced by: islpln2 38920 llnmlplnN 38923 lplnnle2at 38925 lplnneat 38929 lplnnelln 38930 llncvrlpln2 38941 2lplnmN 38943 lplncmp 38946 lplnexatN 38947 lplnexllnN 38948 2llnjaN 38950 islvol3 38960 lvoli3 38961 lvolnle3at 38966 lplncvrlvol2 38999 lplncvrlvol 39000 lvolcmp 39001 2lplnm2N 39005 2lplnmj 39006 dalemyeb 39033 dalem10 39057 dalem16 39063 dalem44 39100 dalem55 39111 |
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