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Theorem lvolbase 36594
Description: A 3-dim lattice volume is a lattice element. (Contributed by NM, 1-Jul-2012.)
Hypotheses
Ref Expression
lvolbase.b 𝐵 = (Base‘𝐾)
lvolbase.v 𝑉 = (LVols‘𝐾)
Assertion
Ref Expression
lvolbase (𝑋𝑉𝑋𝐵)

Proof of Theorem lvolbase
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 n0i 4296 . . . 4 (𝑋𝑉 → ¬ 𝑉 = ∅)
2 lvolbase.v . . . . 5 𝑉 = (LVols‘𝐾)
32eqeq1i 2823 . . . 4 (𝑉 = ∅ ↔ (LVols‘𝐾) = ∅)
41, 3sylnib 329 . . 3 (𝑋𝑉 → ¬ (LVols‘𝐾) = ∅)
5 fvprc 6656 . . 3 𝐾 ∈ V → (LVols‘𝐾) = ∅)
64, 5nsyl2 143 . 2 (𝑋𝑉𝐾 ∈ V)
7 lvolbase.b . . . 4 𝐵 = (Base‘𝐾)
8 eqid 2818 . . . 4 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
9 eqid 2818 . . . 4 (LPlanes‘𝐾) = (LPlanes‘𝐾)
107, 8, 9, 2islvol 36589 . . 3 (𝐾 ∈ V → (𝑋𝑉 ↔ (𝑋𝐵 ∧ ∃𝑥 ∈ (LPlanes‘𝐾)𝑥( ⋖ ‘𝐾)𝑋)))
1110simprbda 499 . 2 ((𝐾 ∈ V ∧ 𝑋𝑉) → 𝑋𝐵)
126, 11mpancom 684 1 (𝑋𝑉𝑋𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wrex 3136  Vcvv 3492  c0 4288   class class class wbr 5057  cfv 6348  Basecbs 16471  ccvr 36278  LPlanesclpl 36508  LVolsclvol 36509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-lvols 36516
This theorem is referenced by:  islvol2  36596  lvolnle3at  36598  lvolneatN  36604  lvolnelln  36605  lvolnelpln  36606  lplncvrlvol2  36631  lvolcmp  36633  2lplnja  36635
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