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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islvol4 | Structured version Visualization version GIF version |
Description: The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.) |
Ref | Expression |
---|---|
lvolset.b | β’ π΅ = (BaseβπΎ) |
lvolset.c | β’ πΆ = ( β βπΎ) |
lvolset.p | β’ π = (LPlanesβπΎ) |
lvolset.v | β’ π = (LVolsβπΎ) |
Ref | Expression |
---|---|
islvol4 | β’ ((πΎ β π΄ β§ π β π΅) β (π β π β βπ¦ β π π¦πΆπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvolset.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | lvolset.c | . . 3 β’ πΆ = ( β βπΎ) | |
3 | lvolset.p | . . 3 β’ π = (LPlanesβπΎ) | |
4 | lvolset.v | . . 3 β’ π = (LVolsβπΎ) | |
5 | 1, 2, 3, 4 | islvol 38956 | . 2 β’ (πΎ β π΄ β (π β π β (π β π΅ β§ βπ¦ β π π¦πΆπ))) |
6 | 5 | baibd 539 | 1 β’ ((πΎ β π΄ β§ π β π΅) β (π β π β βπ¦ β π π¦πΆπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βwrex 3064 class class class wbr 5141 βcfv 6536 Basecbs 17150 β ccvr 38644 LPlanesclpl 38875 LVolsclvol 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 6488 df-fun 6538 df-fv 6544 df-lvols 38883 |
This theorem is referenced by: islvol3 38959 lvolcmp 39000 |
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