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Theorem islvol4 36819
Description: The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)
Hypotheses
Ref Expression
lvolset.b 𝐵 = (Base‘𝐾)
lvolset.c 𝐶 = ( ⋖ ‘𝐾)
lvolset.p 𝑃 = (LPlanes‘𝐾)
lvolset.v 𝑉 = (LVols‘𝐾)
Assertion
Ref Expression
islvol4 ((𝐾𝐴𝑋𝐵) → (𝑋𝑉 ↔ ∃𝑦𝑃 𝑦𝐶𝑋))
Distinct variable groups:   𝑦,𝑃   𝑦,𝐾   𝑦,𝑋
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦)   𝐶(𝑦)   𝑉(𝑦)

Proof of Theorem islvol4
StepHypRef Expression
1 lvolset.b . . 3 𝐵 = (Base‘𝐾)
2 lvolset.c . . 3 𝐶 = ( ⋖ ‘𝐾)
3 lvolset.p . . 3 𝑃 = (LPlanes‘𝐾)
4 lvolset.v . . 3 𝑉 = (LVols‘𝐾)
51, 2, 3, 4islvol 36818 . 2 (𝐾𝐴 → (𝑋𝑉 ↔ (𝑋𝐵 ∧ ∃𝑦𝑃 𝑦𝐶𝑋)))
65baibd 543 1 ((𝐾𝐴𝑋𝐵) → (𝑋𝑉 ↔ ∃𝑦𝑃 𝑦𝐶𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wrex 3134   class class class wbr 5052  cfv 6343  Basecbs 16483  ccvr 36507  LPlanesclpl 36737  LVolsclvol 36738
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 5189  ax-nul 5196  ax-pr 5317
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 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-lvols 36745
This theorem is referenced by:  islvol3  36821  lvolcmp  36862
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