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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 39078 | . 2 β’ (πΎ β π΄ β (π β π β (π β π΅ β§ βπ¦ β π π¦πΆπ))) |
6 | 5 | baibd 538 | 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 394 = wceq 1533 β wcel 2098 βwrex 3067 class class class wbr 5152 βcfv 6553 Basecbs 17187 β ccvr 38766 LPlanesclpl 38997 LVolsclvol 38998 |
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 39005 |
This theorem is referenced by: islvol3 39081 lvolcmp 39122 |
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