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Theorem lplnset 29718
Description: The set of lattice planes in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)
Hypotheses
Ref Expression
lplnset.b  |-  B  =  ( Base `  K
)
lplnset.c  |-  C  =  (  <o  `  K )
lplnset.n  |-  N  =  ( LLines `  K )
lplnset.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
lplnset  |-  ( K  e.  A  ->  P  =  { x  e.  B  |  E. y  e.  N  y C x } )
Distinct variable groups:    y, N    x, B    x, y, K
Allowed substitution hints:    A( x, y)    B( y)    C( x, y)    P( x, y)    N( x)

Proof of Theorem lplnset
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 lplnset.p . . 3  |-  P  =  ( LPlanes `  K )
3 fveq2 5525 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 lplnset.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2333 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 fveq2 5525 . . . . . . 7  |-  ( k  =  K  ->  ( LLines `
 k )  =  ( LLines `  K )
)
7 lplnset.n . . . . . . 7  |-  N  =  ( LLines `  K )
86, 7syl6eqr 2333 . . . . . 6  |-  ( k  =  K  ->  ( LLines `
 k )  =  N )
9 fveq2 5525 . . . . . . . 8  |-  ( k  =  K  ->  (  <o  `  k )  =  (  <o  `  K )
)
10 lplnset.c . . . . . . . 8  |-  C  =  (  <o  `  K )
119, 10syl6eqr 2333 . . . . . . 7  |-  ( k  =  K  ->  (  <o  `  k )  =  C )
1211breqd 4034 . . . . . 6  |-  ( k  =  K  ->  (
y (  <o  `  k
) x  <->  y C x ) )
138, 12rexeqbidv 2749 . . . . 5  |-  ( k  =  K  ->  ( E. y  e.  ( LLines `
 k ) y (  <o  `  k )
x  <->  E. y  e.  N  y C x ) )
145, 13rabeqbidv 2783 . . . 4  |-  ( k  =  K  ->  { x  e.  ( Base `  k
)  |  E. y  e.  ( LLines `  k )
y (  <o  `  k
) x }  =  { x  e.  B  |  E. y  e.  N  y C x } )
15 df-lplanes 29688 . . . 4  |-  LPlanes  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. y  e.  ( LLines `  k ) y ( 
<o  `  k ) x } )
16 fvex 5539 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2353 . . . . 5  |-  B  e. 
_V
1817rabex 4165 . . . 4  |-  { x  e.  B  |  E. y  e.  N  y C x }  e.  _V
1914, 15, 18fvmpt 5602 . . 3  |-  ( K  e.  _V  ->  ( LPlanes
`  K )  =  { x  e.  B  |  E. y  e.  N  y C x } )
202, 19syl5eq 2327 . 2  |-  ( K  e.  _V  ->  P  =  { x  e.  B  |  E. y  e.  N  y C x } )
211, 20syl 15 1  |-  ( K  e.  A  ->  P  =  { x  e.  B  |  E. y  e.  N  y C x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788   class class class wbr 4023   ` cfv 5255   Basecbs 13148    <o ccvr 29452   LLinesclln 29680   LPlanesclpl 29681
This theorem is referenced by:  islpln  29719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-lplanes 29688
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