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Theorem lvolset 29686
Description: The set of 3-dim lattice volumes in a Hilbert lattice. (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
lvolset  |-  ( K  e.  A  ->  V  =  { x  e.  B  |  E. y  e.  P  y C x } )
Distinct variable groups:    y, P    x, B    x, y, K
Allowed substitution hints:    A( x, y)    B( y)    C( x, y)    P( x)    V( x, y)

Proof of Theorem lvolset
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2907 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 lvolset.v . . 3  |-  V  =  ( LVols `  K )
3 fveq2 5668 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 lvolset.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2437 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 fveq2 5668 . . . . . . 7  |-  ( k  =  K  ->  ( LPlanes
`  k )  =  ( LPlanes `  K )
)
7 lvolset.p . . . . . . 7  |-  P  =  ( LPlanes `  K )
86, 7syl6eqr 2437 . . . . . 6  |-  ( k  =  K  ->  ( LPlanes
`  k )  =  P )
9 fveq2 5668 . . . . . . . 8  |-  ( k  =  K  ->  (  <o  `  k )  =  (  <o  `  K )
)
10 lvolset.c . . . . . . . 8  |-  C  =  (  <o  `  K )
119, 10syl6eqr 2437 . . . . . . 7  |-  ( k  =  K  ->  (  <o  `  k )  =  C )
1211breqd 4164 . . . . . 6  |-  ( k  =  K  ->  (
y (  <o  `  k
) x  <->  y C x ) )
138, 12rexeqbidv 2860 . . . . 5  |-  ( k  =  K  ->  ( E. y  e.  ( LPlanes
`  k ) y (  <o  `  k )
x  <->  E. y  e.  P  y C x ) )
145, 13rabeqbidv 2894 . . . 4  |-  ( k  =  K  ->  { x  e.  ( Base `  k
)  |  E. y  e.  ( LPlanes `  k )
y (  <o  `  k
) x }  =  { x  e.  B  |  E. y  e.  P  y C x } )
15 df-lvols 29614 . . . 4  |-  LVols  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. y  e.  ( LPlanes `  k ) y ( 
<o  `  k ) x } )
16 fvex 5682 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2457 . . . . 5  |-  B  e. 
_V
1817rabex 4295 . . . 4  |-  { x  e.  B  |  E. y  e.  P  y C x }  e.  _V
1914, 15, 18fvmpt 5745 . . 3  |-  ( K  e.  _V  ->  ( LVols `  K )  =  { x  e.  B  |  E. y  e.  P  y C x } )
202, 19syl5eq 2431 . 2  |-  ( K  e.  _V  ->  V  =  { x  e.  B  |  E. y  e.  P  y C x } )
211, 20syl 16 1  |-  ( K  e.  A  ->  V  =  { x  e.  B  |  E. y  e.  P  y C x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   E.wrex 2650   {crab 2653   _Vcvv 2899   class class class wbr 4153   ` cfv 5394   Basecbs 13396    <o ccvr 29377   LPlanesclpl 29606   LVolsclvol 29607
This theorem is referenced by:  islvol  29687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-lvols 29614
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