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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 4337 | . . . 4 β’ (π β π β Β¬ π = β ) | |
2 | lplnbase.p | . . . . 5 β’ π = (LPlanesβπΎ) | |
3 | 2 | eqeq1i 2733 | . . . 4 β’ (π = β β (LPlanesβπΎ) = β ) |
4 | 1, 3 | sylnib 327 | . . 3 β’ (π β π β Β¬ (LPlanesβπΎ) = β ) |
5 | fvprc 6894 | . . 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 2728 | . . . 4 β’ ( β βπΎ) = ( β βπΎ) | |
9 | eqid 2728 | . . . 4 β’ (LLinesβπΎ) = (LLinesβπΎ) | |
10 | 7, 8, 9, 2 | islpln 39043 | . . 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 686 | 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 3067 Vcvv 3473 β c0 4326 class class class wbr 5152 βcfv 6553 Basecbs 17189 β ccvr 38774 LLinesclln 39004 LPlanesclpl 39005 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-lplanes 39012 |
This theorem is referenced by: islpln2 39049 llnmlplnN 39052 lplnnle2at 39054 lplnneat 39058 lplnnelln 39059 llncvrlpln2 39070 2lplnmN 39072 lplncmp 39075 lplnexatN 39076 lplnexllnN 39077 2llnjaN 39079 islvol3 39089 lvoli3 39090 lvolnle3at 39095 lplncvrlvol2 39128 lplncvrlvol 39129 lvolcmp 39130 2lplnm2N 39134 2lplnmj 39135 dalemyeb 39162 dalem10 39186 dalem16 39192 dalem44 39229 dalem55 39240 |
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