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Theorem lcfls1N 32395
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 496 . 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 32394 . . 3  |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G )  /\  (  ._|_  `  ( L `  G )
)  C_  Q )
)
5 3anass 941 . . 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 242 . 2  |-  ( G  e.  C  <->  ( G  e.  F  /\  (
(  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =  ( L `  G
)  /\  (  ._|_  `  ( L `  G
) )  C_  Q
) ) )
72, 6syl6rbbr 257 1  |-  ( ph  ->  ( G  e.  C  <->  ( (  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =  ( L `  G
)  /\  (  ._|_  `  ( L `  G
) )  C_  Q
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {crab 2711    C_ wss 3322   ` cfv 5456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464
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