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Theorem dilsetN 30342
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 30341 . . 3  |-  ( K  e.  B  ->  L  =  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `  d
)  ->  ( f `  x )  =  x ) } ) )
76fveq1d 5527 . 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 5525 . . . . . . 7  |-  ( d  =  D  ->  ( W `  d )  =  ( W `  D ) )
98sseq2d 3206 . . . . . 6  |-  ( d  =  D  ->  (
x  C_  ( W `  d )  <->  x  C_  ( W `  D )
) )
109imbi1d 308 . . . . 5  |-  ( d  =  D  ->  (
( x  C_  ( W `  d )  ->  ( f `  x
)  =  x )  <-> 
( x  C_  ( W `  D )  ->  ( f `  x
)  =  x ) ) )
1110ralbidv 2563 . . . 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 2780 . . 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 2283 . . 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 5539 . . . . 5  |-  ( PAut `  K )  e.  _V
154, 14eqeltri 2353 . . . 4  |-  M  e. 
_V
1615rabex 4165 . . 3  |-  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  D )  ->  (
f `  x )  =  x ) }  e.  _V
1712, 13, 16fvmpt 5602 . 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 2335 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152    e. cmpt 4077   ` cfv 5255   Atomscatm 29453   PSubSpcpsubsp 29685   WAtomscwpointsN 30175   PAutcpautN 30176   DilcdilN 30291
This theorem is referenced by:  isdilN  30343
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-dilN 30295
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