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Theorem dilfsetN 35978
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 3118 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 dilset.l . . 3  |-  L  =  ( Dil `  K
)
3 fveq2 5872 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 dilset.a . . . . . 6  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2516 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5872 . . . . . . 7  |-  ( k  =  K  ->  ( PAut `  k )  =  ( PAut `  K
) )
7 dilset.m . . . . . . 7  |-  M  =  ( PAut `  K
)
86, 7syl6eqr 2516 . . . . . 6  |-  ( k  =  K  ->  ( PAut `  k )  =  M )
9 fveq2 5872 . . . . . . . 8  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  ( PSubSp `  K )
)
10 dilset.s . . . . . . . 8  |-  S  =  ( PSubSp `  K )
119, 10syl6eqr 2516 . . . . . . 7  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  S )
12 fveq2 5872 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  ( WAtoms `  K )
)
13 dilset.w . . . . . . . . . . 11  |-  W  =  ( WAtoms `  K )
1412, 13syl6eqr 2516 . . . . . . . . . 10  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  W )
1514fveq1d 5874 . . . . . . . . 9  |-  ( k  =  K  ->  (
( WAtoms `  k ) `  d )  =  ( W `  d ) )
1615sseq2d 3527 . . . . . . . 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 3068 . . . . . 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 3104 . . . . 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 4535 . . . 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 35931 . . . 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 5882 . . . . . 6  |-  ( Atoms `  K )  e.  _V
234, 22eqeltri 2541 . . . . 5  |-  A  e. 
_V
2423mptex 6144 . . . 4  |-  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  d )  ->  (
f `  x )  =  x ) } )  e.  _V
2520, 21, 24fvmpt 5956 . . 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 2510 . 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 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109    C_ wss 3471    |-> cmpt 4515   ` cfv 5594   Atomscatm 35089   PSubSpcpsubsp 35321   WAtomscwpointsN 35811   PAutcpautN 35812   DilcdilN 35927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-dilN 35931
This theorem is referenced by:  dilsetN  35979
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