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Theorem lplnset 33269
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 3002 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 lplnset.p . . 3  |-  P  =  ( LPlanes `  K )
3 fveq2 5712 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 lplnset.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2493 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 fveq2 5712 . . . . . . 7  |-  ( k  =  K  ->  ( LLines `
 k )  =  ( LLines `  K )
)
7 lplnset.n . . . . . . 7  |-  N  =  ( LLines `  K )
86, 7syl6eqr 2493 . . . . . 6  |-  ( k  =  K  ->  ( LLines `
 k )  =  N )
9 fveq2 5712 . . . . . . . 8  |-  ( k  =  K  ->  (  <o  `  k )  =  (  <o  `  K )
)
10 lplnset.c . . . . . . . 8  |-  C  =  (  <o  `  K )
119, 10syl6eqr 2493 . . . . . . 7  |-  ( k  =  K  ->  (  <o  `  k )  =  C )
1211breqd 4324 . . . . . 6  |-  ( k  =  K  ->  (
y (  <o  `  k
) x  <->  y C x ) )
138, 12rexeqbidv 2953 . . . . 5  |-  ( k  =  K  ->  ( E. y  e.  ( LLines `
 k ) y (  <o  `  k )
x  <->  E. y  e.  N  y C x ) )
145, 13rabeqbidv 2988 . . . 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 33239 . . . 4  |-  LPlanes  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. y  e.  ( LLines `  k ) y ( 
<o  `  k ) x } )
16 fvex 5722 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2513 . . . . 5  |-  B  e. 
_V
1817rabex 4464 . . . 4  |-  { x  e.  B  |  E. y  e.  N  y C x }  e.  _V
1914, 15, 18fvmpt 5795 . . 3  |-  ( K  e.  _V  ->  ( LPlanes
`  K )  =  { x  e.  B  |  E. y  e.  N  y C x } )
202, 19syl5eq 2487 . 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 1369    e. wcel 1756   E.wrex 2737   {crab 2740   _Vcvv 2993   class class class wbr 4313   ` cfv 5439   Basecbs 14195    <o ccvr 33003   LLinesclln 33231   LPlanesclpl 33232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-iota 5402  df-fun 5441  df-fv 5447  df-lplanes 33239
This theorem is referenced by:  islpln  33270
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