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Theorem lplnset 35650
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 3115 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 lplnset.p . . 3  |-  P  =  ( LPlanes `  K )
3 fveq2 5848 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 lplnset.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2513 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 fveq2 5848 . . . . . . 7  |-  ( k  =  K  ->  ( LLines `
 k )  =  ( LLines `  K )
)
7 lplnset.n . . . . . . 7  |-  N  =  ( LLines `  K )
86, 7syl6eqr 2513 . . . . . 6  |-  ( k  =  K  ->  ( LLines `
 k )  =  N )
9 fveq2 5848 . . . . . . . 8  |-  ( k  =  K  ->  (  <o  `  k )  =  (  <o  `  K )
)
10 lplnset.c . . . . . . . 8  |-  C  =  (  <o  `  K )
119, 10syl6eqr 2513 . . . . . . 7  |-  ( k  =  K  ->  (  <o  `  k )  =  C )
1211breqd 4450 . . . . . 6  |-  ( k  =  K  ->  (
y (  <o  `  k
) x  <->  y C x ) )
138, 12rexeqbidv 3066 . . . . 5  |-  ( k  =  K  ->  ( E. y  e.  ( LLines `
 k ) y (  <o  `  k )
x  <->  E. y  e.  N  y C x ) )
145, 13rabeqbidv 3101 . . . 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 35620 . . . 4  |-  LPlanes  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. y  e.  ( LLines `  k ) y ( 
<o  `  k ) x } )
16 fvex 5858 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2538 . . . . 5  |-  B  e. 
_V
1817rabex 4588 . . . 4  |-  { x  e.  B  |  E. y  e.  N  y C x }  e.  _V
1914, 15, 18fvmpt 5931 . . 3  |-  ( K  e.  _V  ->  ( LPlanes
`  K )  =  { x  e.  B  |  E. y  e.  N  y C x } )
202, 19syl5eq 2507 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   E.wrex 2805   {crab 2808   _Vcvv 3106   class class class wbr 4439   ` cfv 5570   Basecbs 14716    <o ccvr 35384   LLinesclln 35612   LPlanesclpl 35613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-lplanes 35620
This theorem is referenced by:  islpln  35651
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