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Theorem dilfsetN 30317
Description: The mapping from fiducial atom to set of dilations. (Contributed by NM, 30-Jan-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
dilfsetN  |-  ( K  e.  B  ->  L  =  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `  d
)  ->  ( f `  x )  =  x ) } ) )
Distinct variable groups:    A, d    f, d, x, K    f, M    x, S
Allowed substitution hints:    A( x, f)    B( x, f, d)    S( f, d)    L( x, f, d)    M( x, d)    W( x, f, d)

Proof of Theorem dilfsetN
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2900 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 dilset.l . . 3  |-  L  =  ( Dil `  K
)
3 fveq2 5661 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 dilset.a . . . . . 6  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2430 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5661 . . . . . . 7  |-  ( k  =  K  ->  ( PAut `  k )  =  ( PAut `  K
) )
7 dilset.m . . . . . . 7  |-  M  =  ( PAut `  K
)
86, 7syl6eqr 2430 . . . . . 6  |-  ( k  =  K  ->  ( PAut `  k )  =  M )
9 fveq2 5661 . . . . . . . 8  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  ( PSubSp `  K )
)
10 dilset.s . . . . . . . 8  |-  S  =  ( PSubSp `  K )
119, 10syl6eqr 2430 . . . . . . 7  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  S )
12 fveq2 5661 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  ( WAtoms `  K )
)
13 dilset.w . . . . . . . . . . 11  |-  W  =  ( WAtoms `  K )
1412, 13syl6eqr 2430 . . . . . . . . . 10  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  W )
1514fveq1d 5663 . . . . . . . . 9  |-  ( k  =  K  ->  (
( WAtoms `  k ) `  d )  =  ( W `  d ) )
1615sseq2d 3312 . . . . . . . 8  |-  ( k  =  K  ->  (
x  C_  ( ( WAtoms `
 k ) `  d )  <->  x  C_  ( W `  d )
) )
1716imbi1d 309 . . . . . . 7  |-  ( k  =  K  ->  (
( x  C_  (
( WAtoms `  k ) `  d )  ->  (
f `  x )  =  x )  <->  ( x  C_  ( W `  d
)  ->  ( f `  x )  =  x ) ) )
1811, 17raleqbidv 2852 . . . . . 6  |-  ( k  =  K  ->  ( A. x  e.  ( PSubSp `
 k ) ( x  C_  ( ( WAtoms `
 k ) `  d )  ->  (
f `  x )  =  x )  <->  A. x  e.  S  ( x  C_  ( W `  d
)  ->  ( f `  x )  =  x ) ) )
198, 18rabeqbidv 2887 . . . . 5  |-  ( k  =  K  ->  { f  e.  ( PAut `  k
)  |  A. x  e.  ( PSubSp `  k )
( x  C_  (
( WAtoms `  k ) `  d )  ->  (
f `  x )  =  x ) }  =  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `
 d )  -> 
( f `  x
)  =  x ) } )
205, 19mpteq12dv 4221 . . . 4  |-  ( k  =  K  ->  (
d  e.  ( Atoms `  k )  |->  { f  e.  ( PAut `  k
)  |  A. x  e.  ( PSubSp `  k )
( x  C_  (
( WAtoms `  k ) `  d )  ->  (
f `  x )  =  x ) } )  =  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `  d
)  ->  ( f `  x )  =  x ) } ) )
21 df-dilN 30271 . . . 4  |-  Dil  =  ( k  e.  _V  |->  ( d  e.  (
Atoms `  k )  |->  { f  e.  ( PAut `  k )  |  A. x  e.  ( PSubSp `  k ) ( x 
C_  ( ( WAtoms `  k ) `  d
)  ->  ( f `  x )  =  x ) } ) )
22 fvex 5675 . . . . . 6  |-  ( Atoms `  K )  e.  _V
234, 22eqeltri 2450 . . . . 5  |-  A  e. 
_V
2423mptex 5898 . . . 4  |-  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  d )  ->  (
f `  x )  =  x ) } )  e.  _V
2520, 21, 24fvmpt 5738 . . 3  |-  ( K  e.  _V  ->  ( Dil `  K )  =  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `
 d )  -> 
( f `  x
)  =  x ) } ) )
262, 25syl5eq 2424 . 2  |-  ( K  e.  _V  ->  L  =  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `  d
)  ->  ( f `  x )  =  x ) } ) )
271, 26syl 16 1  |-  ( K  e.  B  ->  L  =  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `  d
)  ->  ( f `  x )  =  x ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   A.wral 2642   {crab 2646   _Vcvv 2892    C_ wss 3256    e. cmpt 4200   ` cfv 5387   Atomscatm 29429   PSubSpcpsubsp 29661   WAtomscwpointsN 30151   PAutcpautN 30152   DilcdilN 30267
This theorem is referenced by:  dilsetN  30318
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-dilN 30271
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