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Theorem lvolset 30208
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 2956 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 lvolset.v . . 3  |-  V  =  ( LVols `  K )
3 fveq2 5719 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 lvolset.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2485 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 fveq2 5719 . . . . . . 7  |-  ( k  =  K  ->  ( LPlanes
`  k )  =  ( LPlanes `  K )
)
7 lvolset.p . . . . . . 7  |-  P  =  ( LPlanes `  K )
86, 7syl6eqr 2485 . . . . . 6  |-  ( k  =  K  ->  ( LPlanes
`  k )  =  P )
9 fveq2 5719 . . . . . . . 8  |-  ( k  =  K  ->  (  <o  `  k )  =  (  <o  `  K )
)
10 lvolset.c . . . . . . . 8  |-  C  =  (  <o  `  K )
119, 10syl6eqr 2485 . . . . . . 7  |-  ( k  =  K  ->  (  <o  `  k )  =  C )
1211breqd 4215 . . . . . 6  |-  ( k  =  K  ->  (
y (  <o  `  k
) x  <->  y C x ) )
138, 12rexeqbidv 2909 . . . . 5  |-  ( k  =  K  ->  ( E. y  e.  ( LPlanes
`  k ) y (  <o  `  k )
x  <->  E. y  e.  P  y C x ) )
145, 13rabeqbidv 2943 . . . 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 30136 . . . 4  |-  LVols  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. y  e.  ( LPlanes `  k ) y ( 
<o  `  k ) x } )
16 fvex 5733 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2505 . . . . 5  |-  B  e. 
_V
1817rabex 4346 . . . 4  |-  { x  e.  B  |  E. y  e.  P  y C x }  e.  _V
1914, 15, 18fvmpt 5797 . . 3  |-  ( K  e.  _V  ->  ( LVols `  K )  =  { x  e.  B  |  E. y  e.  P  y C x } )
202, 19syl5eq 2479 . 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 1652    e. wcel 1725   E.wrex 2698   {crab 2701   _Vcvv 2948   class class class wbr 4204   ` cfv 5445   Basecbs 13457    <o ccvr 29899   LPlanesclpl 30128   LVolsclvol 30129
This theorem is referenced by:  islvol  30209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-iota 5409  df-fun 5447  df-fv 5453  df-lvols 30136
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