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Theorem lvolbase 36774
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 4280 . . . 4 (𝑋𝑉 → ¬ 𝑉 = ∅)
2 lvolbase.v . . . . 5 𝑉 = (LVols‘𝐾)
32eqeq1i 2829 . . . 4 (𝑉 = ∅ ↔ (LVols‘𝐾) = ∅)
41, 3sylnib 331 . . 3 (𝑋𝑉 → ¬ (LVols‘𝐾) = ∅)
5 fvprc 6644 . . 3 𝐾 ∈ V → (LVols‘𝐾) = ∅)
64, 5nsyl2 143 . 2 (𝑋𝑉𝐾 ∈ V)
7 lvolbase.b . . . 4 𝐵 = (Base‘𝐾)
8 eqid 2824 . . . 4 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
9 eqid 2824 . . . 4 (LPlanes‘𝐾) = (LPlanes‘𝐾)
107, 8, 9, 2islvol 36769 . . 3 (𝐾 ∈ V → (𝑋𝑉 ↔ (𝑋𝐵 ∧ ∃𝑥 ∈ (LPlanes‘𝐾)𝑥( ⋖ ‘𝐾)𝑋)))
1110simprbda 502 . 2 ((𝐾 ∈ V ∧ 𝑋𝑉) → 𝑋𝐵)
126, 11mpancom 687 1 (𝑋𝑉𝑋𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wrex 3133  Vcvv 3479  c0 4274   class class class wbr 5047  cfv 6336  Basecbs 16472  ccvr 36458  LPlanesclpl 36688  LVolsclvol 36689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6295  df-fun 6338  df-fv 6344  df-lvols 36696
This theorem is referenced by:  islvol2  36776  lvolnle3at  36778  lvolneatN  36784  lvolnelln  36785  lvolnelpln  36786  lplncvrlvol2  36811  lvolcmp  36813  2lplnja  36815
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