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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvolbase | Structured version Visualization version GIF version |
Description: A 3-dim lattice volume is a lattice element. (Contributed by NM, 1-Jul-2012.) |
Ref | Expression |
---|---|
lvolbase.b | β’ π΅ = (BaseβπΎ) |
lvolbase.v | β’ π = (LVolsβπΎ) |
Ref | Expression |
---|---|
lvolbase | 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 | lvolbase.v | . . . . 5 β’ π = (LVolsβπΎ) | |
3 | 2 | eqeq1i 2733 | . . . 4 β’ (π = β β (LVolsβπΎ) = β ) |
4 | 1, 3 | sylnib 327 | . . 3 β’ (π β π β Β¬ (LVolsβπΎ) = β ) |
5 | fvprc 6894 | . . 3 β’ (Β¬ πΎ β V β (LVolsβπΎ) = β ) | |
6 | 4, 5 | nsyl2 141 | . 2 β’ (π β π β πΎ β V) |
7 | lvolbase.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
8 | eqid 2728 | . . . 4 β’ ( β βπΎ) = ( β βπΎ) | |
9 | eqid 2728 | . . . 4 β’ (LPlanesβπΎ) = (LPlanesβπΎ) | |
10 | 7, 8, 9, 2 | islvol 39086 | . . 3 β’ (πΎ β V β (π β π β (π β π΅ β§ βπ₯ β (LPlanesβπΎ)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 LPlanesclpl 39005 LVolsclvol 39006 |
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-lvols 39013 |
This theorem is referenced by: islvol2 39093 lvolnle3at 39095 lvolneatN 39101 lvolnelln 39102 lvolnelpln 39103 lplncvrlvol2 39128 lvolcmp 39130 2lplnja 39132 |
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