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Theorem islvol 33580
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  =  (  <o  `  K )
lvolset.p  |-  P  =  ( LPlanes `  K )
lvolset.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
islvol  |-  ( K  e.  A  ->  ( X  e.  V  <->  ( X  e.  B  /\  E. y  e.  P  y C X ) ) )
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  =  (  <o  `  K )
3 lvolset.p . . . 4  |-  P  =  ( LPlanes `  K )
4 lvolset.v . . . 4  |-  V  =  ( LVols `  K )
51, 2, 3, 4lvolset 33579 . . 3  |-  ( K  e.  A  ->  V  =  { x  e.  B  |  E. y  e.  P  y C x } )
65eleq2d 2524 . 2  |-  ( K  e.  A  ->  ( X  e.  V  <->  X  e.  { x  e.  B  |  E. y  e.  P  y C x } ) )
7 breq2 4407 . . . 4  |-  ( x  =  X  ->  (
y C x  <->  y C X ) )
87rexbidv 2868 . . 3  |-  ( x  =  X  ->  ( E. y  e.  P  y C x  <->  E. y  e.  P  y C X ) )
98elrab 3224 . 2  |-  ( X  e.  { x  e.  B  |  E. y  e.  P  y C x }  <->  ( X  e.  B  /\  E. y  e.  P  y C X ) )
106, 9syl6bb 261 1  |-  ( K  e.  A  ->  ( X  e.  V  <->  ( X  e.  B  /\  E. y  e.  P  y C X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2800   {crab 2803   class class class wbr 4403   ` cfv 5529   Basecbs 14296    <o ccvr 33270   LPlanesclpl 33499   LVolsclvol 33500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-lvols 33507
This theorem is referenced by:  islvol4  33581  lvoli  33582  lvolbase  33585  lvolnle3at  33589
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