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Theorem isclm 18578
Description: A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
isclm.f  |-  F  =  (Scalar `  W )
isclm.k  |-  K  =  ( Base `  F
)
Assertion
Ref Expression
isclm  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) )

Proof of Theorem isclm
Dummy variables  f 
k  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5555 . . . . 5  |-  (Scalar `  w )  e.  _V
21a1i 10 . . . 4  |-  ( w  =  W  ->  (Scalar `  w )  e.  _V )
3 fvex 5555 . . . . . 6  |-  ( Base `  f )  e.  _V
43a1i 10 . . . . 5  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  e. 
_V )
5 id 19 . . . . . . . . 9  |-  ( f  =  (Scalar `  w
)  ->  f  =  (Scalar `  w ) )
6 fveq2 5541 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
7 isclm.f . . . . . . . . . 10  |-  F  =  (Scalar `  W )
86, 7syl6eqr 2346 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
95, 8sylan9eqr 2350 . . . . . . . 8  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  f  =  F )
109adantr 451 . . . . . . 7  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
f  =  F )
11 id 19 . . . . . . . . 9  |-  ( k  =  ( Base `  f
)  ->  k  =  ( Base `  f )
)
129fveq2d 5545 . . . . . . . . . 10  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  ( Base `  F
) )
13 isclm.k . . . . . . . . . 10  |-  K  =  ( Base `  F
)
1412, 13syl6eqr 2346 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  K )
1511, 14sylan9eqr 2350 . . . . . . . 8  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
k  =  K )
1615oveq2d 5890 . . . . . . 7  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(flds  k
)  =  (flds  K ) )
1710, 16eqeq12d 2310 . . . . . 6  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( f  =  (flds  k )  <-> 
F  =  (flds  K ) ) )
1815eleq1d 2362 . . . . . 6  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( k  e.  (SubRing ` fld ) 
<->  K  e.  (SubRing ` fld ) ) )
1917, 18anbi12d 691 . . . . 5  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( ( f  =  (flds  k )  /\  k  e.  (SubRing ` fld ) )  <->  ( F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) ) )
204, 19sbcied 3040 . . . 4  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( [. ( Base `  f
)  /  k ]. ( f  =  (flds  k )  /\  k  e.  (SubRing ` fld ) )  <->  ( F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) ) )
212, 20sbcied 3040 . . 3  |-  ( w  =  W  ->  ( [. (Scalar `  w )  /  f ]. [. ( Base `  f )  / 
k ]. ( f  =  (flds  k )  /\  k  e.  (SubRing ` fld ) )  <->  ( F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) ) )
22 df-clm 18577 . . 3  |- CMod  =  {
w  e.  LMod  |  [. (Scalar `  w )  / 
f ]. [. ( Base `  f )  /  k ]. ( f  =  (flds  k )  /\  k  e.  (SubRing ` fld ) ) }
2321, 22elrab2 2938 . 2  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  ( F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) ) )
24 3anass 938 . 2  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  <->  ( W  e.  LMod  /\  ( F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) ) )
2523, 24bitr4i 243 1  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   [.wsbc 3004   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165  Scalarcsca 13227  SubRingcsubrg 15557   LModclmod 15643  ℂfldccnfld 16393  CModcclm 18576
This theorem is referenced by:  clmsca  18579  clmsubrg  18580  clmlmod  18581  isclmi  18591  lmhmclm  18600  cphclm  18641  tchclm  18678
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  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-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-clm 18577
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