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Theorem islvol 35381
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
islvol (𝐾𝐴 → (𝑋𝑉 ↔ (𝑋𝐵 ∧ ∃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 islvol
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 lvolset.b . . . 4 𝐵 = (Base‘𝐾)
2 lvolset.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
3 lvolset.p . . . 4 𝑃 = (LPlanes‘𝐾)
4 lvolset.v . . . 4 𝑉 = (LVols‘𝐾)
51, 2, 3, 4lvolset 35380 . . 3 (𝐾𝐴𝑉 = {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥})
65eleq2d 2834 . 2 (𝐾𝐴 → (𝑋𝑉𝑋 ∈ {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥}))
7 breq2 4787 . . . 4 (𝑥 = 𝑋 → (𝑦𝐶𝑥𝑦𝐶𝑋))
87rexbidv 3198 . . 3 (𝑥 = 𝑋 → (∃𝑦𝑃 𝑦𝐶𝑥 ↔ ∃𝑦𝑃 𝑦𝐶𝑋))
98elrab 3512 . 2 (𝑋 ∈ {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥} ↔ (𝑋𝐵 ∧ ∃𝑦𝑃 𝑦𝐶𝑋))
106, 9syl6bb 276 1 (𝐾𝐴 → (𝑋𝑉 ↔ (𝑋𝐵 ∧ ∃𝑦𝑃 𝑦𝐶𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1629  wcel 2143  wrex 3060  {crab 3063   class class class wbr 4783  cfv 6030  Basecbs 16070  ccvr 35071  LPlanesclpl 35300  LVolsclvol 35301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pr 5033
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ral 3064  df-rex 3065  df-rab 3068  df-v 3350  df-sbc 3585  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-br 4784  df-opab 4844  df-mpt 4861  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-iota 5993  df-fun 6032  df-fv 6038  df-lvols 35308
This theorem is referenced by:  islvol4  35382  lvoli  35383  lvolbase  35386  lvolnle3at  35390
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