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Theorem lplnset 30326
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 2964 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 lplnset.p . . 3  |-  P  =  ( LPlanes `  K )
3 fveq2 5728 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 lplnset.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2486 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 fveq2 5728 . . . . . . 7  |-  ( k  =  K  ->  ( LLines `
 k )  =  ( LLines `  K )
)
7 lplnset.n . . . . . . 7  |-  N  =  ( LLines `  K )
86, 7syl6eqr 2486 . . . . . 6  |-  ( k  =  K  ->  ( LLines `
 k )  =  N )
9 fveq2 5728 . . . . . . . 8  |-  ( k  =  K  ->  (  <o  `  k )  =  (  <o  `  K )
)
10 lplnset.c . . . . . . . 8  |-  C  =  (  <o  `  K )
119, 10syl6eqr 2486 . . . . . . 7  |-  ( k  =  K  ->  (  <o  `  k )  =  C )
1211breqd 4223 . . . . . 6  |-  ( k  =  K  ->  (
y (  <o  `  k
) x  <->  y C x ) )
138, 12rexeqbidv 2917 . . . . 5  |-  ( k  =  K  ->  ( E. y  e.  ( LLines `
 k ) y (  <o  `  k )
x  <->  E. y  e.  N  y C x ) )
145, 13rabeqbidv 2951 . . . 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 30296 . . . 4  |-  LPlanes  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. y  e.  ( LLines `  k ) y ( 
<o  `  k ) x } )
16 fvex 5742 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2506 . . . . 5  |-  B  e. 
_V
1817rabex 4354 . . . 4  |-  { x  e.  B  |  E. y  e.  N  y C x }  e.  _V
1914, 15, 18fvmpt 5806 . . 3  |-  ( K  e.  _V  ->  ( LPlanes
`  K )  =  { x  e.  B  |  E. y  e.  N  y C x } )
202, 19syl5eq 2480 . 2  |-  ( K  e.  _V  ->  P  =  { x  e.  B  |  E. y  e.  N  y C x } )
211, 20syl 16 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 1652    e. wcel 1725   E.wrex 2706   {crab 2709   _Vcvv 2956   class class class wbr 4212   ` cfv 5454   Basecbs 13469    <o ccvr 30060   LLinesclln 30288   LPlanesclpl 30289
This theorem is referenced by:  islpln  30327
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-lplanes 30296
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