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Theorem dilfsetN 33794
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 2980 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 dilset.l . . 3  |-  L  =  ( Dil `  K
)
3 fveq2 5690 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 dilset.a . . . . . 6  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2492 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5690 . . . . . . 7  |-  ( k  =  K  ->  ( PAut `  k )  =  ( PAut `  K
) )
7 dilset.m . . . . . . 7  |-  M  =  ( PAut `  K
)
86, 7syl6eqr 2492 . . . . . 6  |-  ( k  =  K  ->  ( PAut `  k )  =  M )
9 fveq2 5690 . . . . . . . 8  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  ( PSubSp `  K )
)
10 dilset.s . . . . . . . 8  |-  S  =  ( PSubSp `  K )
119, 10syl6eqr 2492 . . . . . . 7  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  S )
12 fveq2 5690 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  ( WAtoms `  K )
)
13 dilset.w . . . . . . . . . . 11  |-  W  =  ( WAtoms `  K )
1412, 13syl6eqr 2492 . . . . . . . . . 10  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  W )
1514fveq1d 5692 . . . . . . . . 9  |-  ( k  =  K  ->  (
( WAtoms `  k ) `  d )  =  ( W `  d ) )
1615sseq2d 3383 . . . . . . . 8  |-  ( k  =  K  ->  (
x  C_  ( ( WAtoms `
 k ) `  d )  <->  x  C_  ( W `  d )
) )
1716imbi1d 317 . . . . . . 7  |-  ( k  =  K  ->  (
( x  C_  (
( WAtoms `  k ) `  d )  ->  (
f `  x )  =  x )  <->  ( x  C_  ( W `  d
)  ->  ( f `  x )  =  x ) ) )
1811, 17raleqbidv 2930 . . . . . 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 2966 . . . . 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 4369 . . . 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 33748 . . . 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 5700 . . . . . 6  |-  ( Atoms `  K )  e.  _V
234, 22eqeltri 2512 . . . . 5  |-  A  e. 
_V
2423mptex 5947 . . . 4  |-  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  d )  ->  (
f `  x )  =  x ) } )  e.  _V
2520, 21, 24fvmpt 5773 . . 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 2486 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2714   {crab 2718   _Vcvv 2971    C_ wss 3327    e. cmpt 4349   ` cfv 5417   Atomscatm 32906   PSubSpcpsubsp 33138   WAtomscwpointsN 33628   PAutcpautN 33629   DilcdilN 33744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-dilN 33748
This theorem is referenced by:  dilsetN  33795
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