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Theorem lvolset 34769
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 3127 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 lvolset.v . . 3  |-  V  =  ( LVols `  K )
3 fveq2 5872 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 lvolset.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2526 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 fveq2 5872 . . . . . . 7  |-  ( k  =  K  ->  ( LPlanes
`  k )  =  ( LPlanes `  K )
)
7 lvolset.p . . . . . . 7  |-  P  =  ( LPlanes `  K )
86, 7syl6eqr 2526 . . . . . 6  |-  ( k  =  K  ->  ( LPlanes
`  k )  =  P )
9 fveq2 5872 . . . . . . . 8  |-  ( k  =  K  ->  (  <o  `  k )  =  (  <o  `  K )
)
10 lvolset.c . . . . . . . 8  |-  C  =  (  <o  `  K )
119, 10syl6eqr 2526 . . . . . . 7  |-  ( k  =  K  ->  (  <o  `  k )  =  C )
1211breqd 4464 . . . . . 6  |-  ( k  =  K  ->  (
y (  <o  `  k
) x  <->  y C x ) )
138, 12rexeqbidv 3078 . . . . 5  |-  ( k  =  K  ->  ( E. y  e.  ( LPlanes
`  k ) y (  <o  `  k )
x  <->  E. y  e.  P  y C x ) )
145, 13rabeqbidv 3113 . . . 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 34697 . . . 4  |-  LVols  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. y  e.  ( LPlanes `  k ) y ( 
<o  `  k ) x } )
16 fvex 5882 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2551 . . . . 5  |-  B  e. 
_V
1817rabex 4604 . . . 4  |-  { x  e.  B  |  E. y  e.  P  y C x }  e.  _V
1914, 15, 18fvmpt 5957 . . 3  |-  ( K  e.  _V  ->  ( LVols `  K )  =  { x  e.  B  |  E. y  e.  P  y C x } )
202, 19syl5eq 2520 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   E.wrex 2818   {crab 2821   _Vcvv 3118   class class class wbr 4453   ` cfv 5594   Basecbs 14507    <o ccvr 34460   LPlanesclpl 34689   LVolsclvol 34690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-lvols 34697
This theorem is referenced by:  islvol  34770
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