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Theorem isdilN 30888
Description: The predicate "is a dilation". (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
isdilN  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( F  e.  ( L `  D )  <-> 
( F  e.  M  /\  A. x  e.  S  ( x  C_  ( W `
 D )  -> 
( F `  x
)  =  x ) ) ) )
Distinct variable groups:    x, K    x, S    x, D    x, F
Allowed substitution hints:    A( x)    B( x)    L( x)    M( x)    W( x)

Proof of Theorem isdilN
Dummy variable  f 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, 5dilsetN 30887 . . 3  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( L `  D
)  =  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  D )  ->  (
f `  x )  =  x ) } )
76eleq2d 2502 . 2  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( F  e.  ( L `  D )  <-> 
F  e.  { f  e.  M  |  A. x  e.  S  (
x  C_  ( W `  D )  ->  (
f `  x )  =  x ) } ) )
8 fveq1 5719 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
98eqeq1d 2443 . . . . 5  |-  ( f  =  F  ->  (
( f `  x
)  =  x  <->  ( F `  x )  =  x ) )
109imbi2d 308 . . . 4  |-  ( f  =  F  ->  (
( x  C_  ( W `  D )  ->  ( f `  x
)  =  x )  <-> 
( x  C_  ( W `  D )  ->  ( F `  x
)  =  x ) ) )
1110ralbidv 2717 . . 3  |-  ( f  =  F  ->  ( A. x  e.  S  ( x  C_  ( W `
 D )  -> 
( f `  x
)  =  x )  <->  A. x  e.  S  ( x  C_  ( W `
 D )  -> 
( F `  x
)  =  x ) ) )
1211elrab 3084 . 2  |-  ( F  e.  { 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 ) ) )
137, 12syl6bb 253 1  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( F  e.  ( 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    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701    C_ wss 3312   ` cfv 5446   Atomscatm 29998   PSubSpcpsubsp 30230   WAtomscwpointsN 30720   PAutcpautN 30721   DilcdilN 30836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-dilN 30840
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