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Theorem islvol4 39079
Description: The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)
Hypotheses
Ref Expression
lvolset.b 𝐡 = (Baseβ€˜πΎ)
lvolset.c 𝐢 = ( β‹– β€˜πΎ)
lvolset.p 𝑃 = (LPlanesβ€˜πΎ)
lvolset.v 𝑉 = (LVolsβ€˜πΎ)
Assertion
Ref Expression
islvol4 ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∈ 𝑉 ↔ βˆƒπ‘¦ ∈ 𝑃 𝑦𝐢𝑋))
Distinct variable groups:   𝑦,𝑃   𝑦,𝐾   𝑦,𝑋
Allowed substitution hints:   𝐴(𝑦)   𝐡(𝑦)   𝐢(𝑦)   𝑉(𝑦)

Proof of Theorem islvol4
StepHypRef Expression
1 lvolset.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 lvolset.c . . 3 𝐢 = ( β‹– β€˜πΎ)
3 lvolset.p . . 3 𝑃 = (LPlanesβ€˜πΎ)
4 lvolset.v . . 3 𝑉 = (LVolsβ€˜πΎ)
51, 2, 3, 4islvol 39078 . 2 (𝐾 ∈ 𝐴 β†’ (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘¦ ∈ 𝑃 𝑦𝐢𝑋)))
65baibd 538 1 ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∈ 𝑉 ↔ βˆƒπ‘¦ ∈ 𝑃 𝑦𝐢𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3067   class class class wbr 5152  β€˜cfv 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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