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Theorem islvol 39176
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 39175 . . 3 (𝐾𝐴𝑉 = {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥})
65eleq2d 2811 . 2 (𝐾𝐴 → (𝑋𝑉𝑋 ∈ {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥}))
7 breq2 5153 . . . 4 (𝑥 = 𝑋 → (𝑦𝐶𝑥𝑦𝐶𝑋))
87rexbidv 3168 . . 3 (𝑥 = 𝑋 → (∃𝑦𝑃 𝑦𝐶𝑥 ↔ ∃𝑦𝑃 𝑦𝐶𝑋))
98elrab 3679 . 2 (𝑋 ∈ {𝑥𝐵 ∣ ∃𝑦𝑃 𝑦𝐶𝑥} ↔ (𝑋𝐵 ∧ ∃𝑦𝑃 𝑦𝐶𝑋))
106, 9bitrdi 286 1 (𝐾𝐴 → (𝑋𝑉 ↔ (𝑋𝐵 ∧ ∃𝑦𝑃 𝑦𝐶𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wrex 3059  {crab 3418   class class class wbr 5149  cfv 6549  Basecbs 17183  ccvr 38864  LPlanesclpl 39095  LVolsclvol 39096
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 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-lvols 39103
This theorem is referenced by:  islvol4  39177  lvoli  39178  lvolbase  39181  lvolnle3at  39185
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