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Theorem islvol4 39518
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 39517 . 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 1535  wcel 2104  wrex 3066   class class class wbr 5149  cfv 6558  Basecbs 17234  ccvr 39205  LPlanesclpl 39436  LVolsclvol 39437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-iota 6510  df-fun 6560  df-fv 6566  df-lvols 39444
This theorem is referenced by:  islvol3  39520  lvolcmp  39561
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