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Theorem lvolbase 38962
Description: A 3-dim lattice volume is a lattice element. (Contributed by NM, 1-Jul-2012.)
Hypotheses
Ref Expression
lvolbase.b 𝐡 = (Baseβ€˜πΎ)
lvolbase.v 𝑉 = (LVolsβ€˜πΎ)
Assertion
Ref Expression
lvolbase (𝑋 ∈ 𝑉 β†’ 𝑋 ∈ 𝐡)

Proof of Theorem lvolbase
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 n0i 4328 . . . 4 (𝑋 ∈ 𝑉 β†’ Β¬ 𝑉 = βˆ…)
2 lvolbase.v . . . . 5 𝑉 = (LVolsβ€˜πΎ)
32eqeq1i 2731 . . . 4 (𝑉 = βˆ… ↔ (LVolsβ€˜πΎ) = βˆ…)
41, 3sylnib 328 . . 3 (𝑋 ∈ 𝑉 β†’ Β¬ (LVolsβ€˜πΎ) = βˆ…)
5 fvprc 6877 . . 3 (Β¬ 𝐾 ∈ V β†’ (LVolsβ€˜πΎ) = βˆ…)
64, 5nsyl2 141 . 2 (𝑋 ∈ 𝑉 β†’ 𝐾 ∈ V)
7 lvolbase.b . . . 4 𝐡 = (Baseβ€˜πΎ)
8 eqid 2726 . . . 4 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
9 eqid 2726 . . . 4 (LPlanesβ€˜πΎ) = (LPlanesβ€˜πΎ)
107, 8, 9, 2islvol 38957 . . 3 (𝐾 ∈ V β†’ (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘₯ ∈ (LPlanesβ€˜πΎ)π‘₯( β‹– β€˜πΎ)𝑋)))
1110simprbda 498 . 2 ((𝐾 ∈ V ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 ∈ 𝐡)
126, 11mpancom 685 1 (𝑋 ∈ 𝑉 β†’ 𝑋 ∈ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064  Vcvv 3468  βˆ…c0 4317   class class class wbr 5141  β€˜cfv 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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