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Theorem dilfsetN 30963
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 2809 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 dilset.l . . 3  |-  L  =  ( Dil `  K
)
3 fveq2 5541 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 dilset.a . . . . . 6  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2346 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5541 . . . . . . 7  |-  ( k  =  K  ->  ( PAut `  k )  =  ( PAut `  K
) )
7 dilset.m . . . . . . 7  |-  M  =  ( PAut `  K
)
86, 7syl6eqr 2346 . . . . . 6  |-  ( k  =  K  ->  ( PAut `  k )  =  M )
9 fveq2 5541 . . . . . . . 8  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  ( PSubSp `  K )
)
10 dilset.s . . . . . . . 8  |-  S  =  ( PSubSp `  K )
119, 10syl6eqr 2346 . . . . . . 7  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  S )
12 fveq2 5541 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  ( WAtoms `  K )
)
13 dilset.w . . . . . . . . . . 11  |-  W  =  ( WAtoms `  K )
1412, 13syl6eqr 2346 . . . . . . . . . 10  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  W )
1514fveq1d 5543 . . . . . . . . 9  |-  ( k  =  K  ->  (
( WAtoms `  k ) `  d )  =  ( W `  d ) )
1615sseq2d 3219 . . . . . . . 8  |-  ( k  =  K  ->  (
x  C_  ( ( WAtoms `
 k ) `  d )  <->  x  C_  ( W `  d )
) )
1716imbi1d 308 . . . . . . 7  |-  ( k  =  K  ->  (
( x  C_  (
( WAtoms `  k ) `  d )  ->  (
f `  x )  =  x )  <->  ( x  C_  ( W `  d
)  ->  ( f `  x )  =  x ) ) )
1811, 17raleqbidv 2761 . . . . . 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 2796 . . . . 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 4114 . . . 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 30917 . . . 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 5555 . . . . . 6  |-  ( Atoms `  K )  e.  _V
234, 22eqeltri 2366 . . . . 5  |-  A  e. 
_V
2423mptex 5762 . . . 4  |-  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  d )  ->  (
f `  x )  =  x ) } )  e.  _V
2520, 21, 24fvmpt 5618 . . 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 2340 . 2  |-  ( K  e.  _V  ->  L  =  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `  d
)  ->  ( f `  x )  =  x ) } ) )
271, 26syl 15 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 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    C_ wss 3165    e. cmpt 4093   ` cfv 5271   Atomscatm 30075   PSubSpcpsubsp 30307   WAtomscwpointsN 30797   PAutcpautN 30798   DilcdilN 30913
This theorem is referenced by:  dilsetN  30964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-dilN 30917
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