Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcfls1N Unicode version

Theorem lcfls1N 31725
Description: Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
lcfls1.c  |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f )  /\  (  ._|_  `  ( L `  f ) )  C_  Q ) }
lcfls1.g  |-  ( ph  ->  G  e.  F )
Assertion
Ref Expression
lcfls1N  |-  ( ph  ->  ( G  e.  C  <->  ( (  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =  ( L `  G
)  /\  (  ._|_  `  ( L `  G
) )  C_  Q
) ) )
Distinct variable groups:    f, F    f, G    f, L    ._|_ , f    Q, f
Allowed substitution hints:    ph( f)    C( f)

Proof of Theorem lcfls1N
StepHypRef Expression
1 lcfls1.g . . 3  |-  ( ph  ->  G  e.  F )
21biantrurd 494 . 2  |-  ( ph  ->  ( ( (  ._|_  `  (  ._|_  `  ( L `
 G ) ) )  =  ( L `
 G )  /\  (  ._|_  `  ( L `  G ) )  C_  Q )  <->  ( G  e.  F  /\  (
(  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =  ( L `  G
)  /\  (  ._|_  `  ( L `  G
) )  C_  Q
) ) ) )
3 lcfls1.c . . . 4  |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f )  /\  (  ._|_  `  ( L `  f ) )  C_  Q ) }
43lcfls1lem 31724 . . 3  |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G )  /\  (  ._|_  `  ( L `  G )
)  C_  Q )
)
5 3anass 938 . . 3  |-  ( ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =  ( L `  G
)  /\  (  ._|_  `  ( L `  G
) )  C_  Q
)  <->  ( G  e.  F  /\  ( ( 
._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G )  /\  (  ._|_  `  ( L `  G )
)  C_  Q )
) )
64, 5bitri 240 . 2  |-  ( G  e.  C  <->  ( G  e.  F  /\  (
(  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =  ( L `  G
)  /\  (  ._|_  `  ( L `  G
) )  C_  Q
) ) )
72, 6syl6rbbr 255 1  |-  ( ph  ->  ( G  e.  C  <->  ( (  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =  ( L `  G
)  /\  (  ._|_  `  ( L `  G
) )  C_  Q
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   ` cfv 5255
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  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-iota 5219  df-fv 5263
  Copyright terms: Public domain W3C validator