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Theorem dilsetN 30267
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 30266 . . 3  |-  ( K  e.  B  ->  L  =  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `  d
)  ->  ( f `  x )  =  x ) } ) )
76fveq1d 5670 . 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 5668 . . . . . . 7  |-  ( d  =  D  ->  ( W `  d )  =  ( W `  D ) )
98sseq2d 3319 . . . . . 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 2669 . . . 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 2891 . . 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 2387 . . 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 5682 . . . . 5  |-  ( PAut `  K )  e.  _V
154, 14eqeltri 2457 . . . 4  |-  M  e. 
_V
1615rabex 4295 . . 3  |-  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  D )  ->  (
f `  x )  =  x ) }  e.  _V
1712, 13, 16fvmpt 5745 . 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 2439 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 1649    e. wcel 1717   A.wral 2649   {crab 2653   _Vcvv 2899    C_ wss 3263    e. cmpt 4207   ` cfv 5394   Atomscatm 29378   PSubSpcpsubsp 29610   WAtomscwpointsN 30100   PAutcpautN 30101   DilcdilN 30216
This theorem is referenced by:  isdilN  30268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-dilN 30220
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