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Theorem lvolset 30443
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 2966 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 lvolset.v . . 3  |-  V  =  ( LVols `  K )
3 fveq2 5731 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 lvolset.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2488 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 fveq2 5731 . . . . . . 7  |-  ( k  =  K  ->  ( LPlanes
`  k )  =  ( LPlanes `  K )
)
7 lvolset.p . . . . . . 7  |-  P  =  ( LPlanes `  K )
86, 7syl6eqr 2488 . . . . . 6  |-  ( k  =  K  ->  ( LPlanes
`  k )  =  P )
9 fveq2 5731 . . . . . . . 8  |-  ( k  =  K  ->  (  <o  `  k )  =  (  <o  `  K )
)
10 lvolset.c . . . . . . . 8  |-  C  =  (  <o  `  K )
119, 10syl6eqr 2488 . . . . . . 7  |-  ( k  =  K  ->  (  <o  `  k )  =  C )
1211breqd 4226 . . . . . 6  |-  ( k  =  K  ->  (
y (  <o  `  k
) x  <->  y C x ) )
138, 12rexeqbidv 2919 . . . . 5  |-  ( k  =  K  ->  ( E. y  e.  ( LPlanes
`  k ) y (  <o  `  k )
x  <->  E. y  e.  P  y C x ) )
145, 13rabeqbidv 2953 . . . 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 30371 . . . 4  |-  LVols  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. y  e.  ( LPlanes `  k ) y ( 
<o  `  k ) x } )
16 fvex 5745 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2508 . . . . 5  |-  B  e. 
_V
1817rabex 4357 . . . 4  |-  { x  e.  B  |  E. y  e.  P  y C x }  e.  _V
1914, 15, 18fvmpt 5809 . . 3  |-  ( K  e.  _V  ->  ( LVols `  K )  =  { x  e.  B  |  E. y  e.  P  y C x } )
202, 19syl5eq 2482 . 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 1653    e. wcel 1726   E.wrex 2708   {crab 2711   _Vcvv 2958   class class class wbr 4215   ` cfv 5457   Basecbs 13474    <o ccvr 30134   LPlanesclpl 30363   LVolsclvol 30364
This theorem is referenced by:  islvol  30444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-lvols 30371
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