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

Theorem dilsetN 30888
Description: The set of dilations for a fiducial atom  D. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
dilset.a  |-  A  =  ( Atoms `  K )
dilset.s  |-  S  =  ( PSubSp `  K )
dilset.w  |-  W  =  ( WAtoms `  K )
dilset.m  |-  M  =  ( PAut `  K
)
dilset.l  |-  L  =  ( Dil `  K
)
Assertion
Ref Expression
dilsetN  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( L `  D
)  =  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  D )  ->  (
f `  x )  =  x ) } )
Distinct variable groups:    x, f, K    f, M    x, S    D, f, x
Allowed substitution hints:    A( x, f)    B( x, f)    S( f)    L( x, f)    M( x)    W( x, f)

Proof of Theorem dilsetN
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 dilset.a . . . 4  |-  A  =  ( Atoms `  K )
2 dilset.s . . . 4  |-  S  =  ( PSubSp `  K )
3 dilset.w . . . 4  |-  W  =  ( WAtoms `  K )
4 dilset.m . . . 4  |-  M  =  ( PAut `  K
)
5 dilset.l . . . 4  |-  L  =  ( Dil `  K
)
61, 2, 3, 4, 5dilfsetN 30887 . . 3  |-  ( K  e.  B  ->  L  =  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `  d
)  ->  ( f `  x )  =  x ) } ) )
76fveq1d 5723 . 2  |-  ( K  e.  B  ->  ( L `  D )  =  ( ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  d )  ->  (
f `  x )  =  x ) } ) `
 D ) )
8 fveq2 5721 . . . . . . 7  |-  ( d  =  D  ->  ( W `  d )  =  ( W `  D ) )
98sseq2d 3369 . . . . . 6  |-  ( d  =  D  ->  (
x  C_  ( W `  d )  <->  x  C_  ( W `  D )
) )
109imbi1d 309 . . . . 5  |-  ( d  =  D  ->  (
( x  C_  ( W `  d )  ->  ( f `  x
)  =  x )  <-> 
( x  C_  ( W `  D )  ->  ( f `  x
)  =  x ) ) )
1110ralbidv 2718 . . . 4  |-  ( d  =  D  ->  ( A. x  e.  S  ( x  C_  ( W `
 d )  -> 
( f `  x
)  =  x )  <->  A. x  e.  S  ( x  C_  ( W `
 D )  -> 
( f `  x
)  =  x ) ) )
1211rabbidv 2941 . . 3  |-  ( d  =  D  ->  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  d )  ->  (
f `  x )  =  x ) }  =  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `
 D )  -> 
( f `  x
)  =  x ) } )
13 eqid 2436 . . 3  |-  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  d )  ->  (
f `  x )  =  x ) } )  =  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `  d
)  ->  ( f `  x )  =  x ) } )
14 fvex 5735 . . . . 5  |-  ( PAut `  K )  e.  _V
154, 14eqeltri 2506 . . . 4  |-  M  e. 
_V
1615rabex 4347 . . 3  |-  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  D )  ->  (
f `  x )  =  x ) }  e.  _V
1712, 13, 16fvmpt 5799 . 2  |-  ( D  e.  A  ->  (
( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `
 d )  -> 
( f `  x
)  =  x ) } ) `  D
)  =  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  D )  ->  (
f `  x )  =  x ) } )
187, 17sylan9eq 2488 1  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( L `  D
)  =  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  D )  ->  (
f `  x )  =  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2698   {crab 2702   _Vcvv 2949    C_ wss 3313    e. cmpt 4259   ` cfv 5447   Atomscatm 29999   PSubSpcpsubsp 30231   WAtomscwpointsN 30721   PAutcpautN 30722   DilcdilN 30837
This theorem is referenced by:  isdilN  30889
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-rep 4313  ax-sep 4323  ax-nul 4331  ax-pr 4396
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 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-dilN 30841
  Copyright terms: Public domain W3C validator