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Theorem islvol4 30385
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
islvol4  |-  ( ( K  e.  A  /\  X  e.  B )  ->  ( X  e.  V  <->  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 islvol4
StepHypRef Expression
1 lvolset.b . . 3  |-  B  =  ( Base `  K
)
2 lvolset.c . . 3  |-  C  =  (  <o  `  K )
3 lvolset.p . . 3  |-  P  =  ( LPlanes `  K )
4 lvolset.v . . 3  |-  V  =  ( LVols `  K )
51, 2, 3, 4islvol 30384 . 2  |-  ( K  e.  A  ->  ( X  e.  V  <->  ( X  e.  B  /\  E. y  e.  P  y C X ) ) )
65baibd 875 1  |-  ( ( K  e.  A  /\  X  e.  B )  ->  ( X  e.  V  <->  E. y  e.  P  y C X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   ` cfv 5271   Basecbs 13164    <o ccvr 30074   LPlanesclpl 30303   LVolsclvol 30304
This theorem is referenced by:  islvol3  30387  lvolcmp  30428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-lvols 30311
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