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Mathbox for Norm Megill |
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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 4328 | . . . 4 β’ (π β π β Β¬ π = β ) | |
2 | lvolbase.v | . . . . 5 β’ π = (LVolsβπΎ) | |
3 | 2 | eqeq1i 2731 | . . . 4 β’ (π = β β (LVolsβπΎ) = β ) |
4 | 1, 3 | sylnib 328 | . . 3 β’ (π β π β Β¬ (LVolsβπΎ) = β ) |
5 | fvprc 6877 | . . 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 2726 | . . . 4 β’ ( β βπΎ) = ( β βπΎ) | |
9 | eqid 2726 | . . . 4 β’ (LPlanesβπΎ) = (LPlanesβπΎ) | |
10 | 7, 8, 9, 2 | islvol 38957 | . . 3 β’ (πΎ β V β (π β π β (π β π΅ β§ βπ₯ β (LPlanesβπΎ)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 LPlanesclpl 38876 LVolsclvol 38877 |
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-lvols 38884 |
This theorem is referenced by: islvol2 38964 lvolnle3at 38966 lvolneatN 38972 lvolnelln 38973 lvolnelpln 38974 lplncvrlvol2 38999 lvolcmp 39001 2lplnja 39003 |
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