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Theorem lvolbase 39091
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 4337 . . . 4 (𝑋 ∈ 𝑉 β†’ Β¬ 𝑉 = βˆ…)
2 lvolbase.v . . . . 5 𝑉 = (LVolsβ€˜πΎ)
32eqeq1i 2733 . . . 4 (𝑉 = βˆ… ↔ (LVolsβ€˜πΎ) = βˆ…)
41, 3sylnib 327 . . 3 (𝑋 ∈ 𝑉 β†’ Β¬ (LVolsβ€˜πΎ) = βˆ…)
5 fvprc 6894 . . 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 2728 . . . 4 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
9 eqid 2728 . . . 4 (LPlanesβ€˜πΎ) = (LPlanesβ€˜πΎ)
107, 8, 9, 2islvol 39086 . . 3 (𝐾 ∈ V β†’ (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘₯ ∈ (LPlanesβ€˜πΎ)π‘₯( β‹– β€˜πΎ)𝑋)))
1110simprbda 497 . 2 ((𝐾 ∈ V ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 ∈ 𝐡)
126, 11mpancom 686 1 (𝑋 ∈ 𝑉 β†’ 𝑋 ∈ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3067  Vcvv 3473  βˆ…c0 4326   class class class wbr 5152  β€˜cfv 6553  Basecbs 17189   β‹– ccvr 38774  LPlanesclpl 39005  LVolsclvol 39006
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 39013
This theorem is referenced by:  islvol2  39093  lvolnle3at  39095  lvolneatN  39101  lvolnelln  39102  lvolnelpln  39103  lplncvrlvol2  39128  lvolcmp  39130  2lplnja  39132
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