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Theorem islvol4 39541
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 39540 . 2 (𝐾𝐴 → (𝑋𝑉 ↔ (𝑋𝐵 ∧ ∃𝑦𝑃 𝑦𝐶𝑋)))
65baibd 539 1 ((𝐾𝐴𝑋𝐵) → (𝑋𝑉 ↔ ∃𝑦𝑃 𝑦𝐶𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2109  wrex 3053   class class class wbr 5102  cfv 6499  Basecbs 17155  ccvr 39228  LPlanesclpl 39459  LVolsclvol 39460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-lvols 39467
This theorem is referenced by:  islvol3  39543  lvolcmp  39584
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