Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  islvol4 Structured version   Visualization version   GIF version

Theorem islvol4 39554
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 39553 . 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 2108  wrex 3069   class class class wbr 5141  cfv 6559  Basecbs 17243  ccvr 39241  LPlanesclpl 39472  LVolsclvol 39473
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-iota 6512  df-fun 6561  df-fv 6567  df-lvols 39480
This theorem is referenced by:  islvol3  39556  lvolcmp  39597
  Copyright terms: Public domain W3C validator