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Theorem isclm 19090
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 5743 . . . . 5  |-  (Scalar `  w )  e.  _V
21a1i 11 . . . 4  |-  ( w  =  W  ->  (Scalar `  w )  e.  _V )
3 fvex 5743 . . . . . 6  |-  ( Base `  f )  e.  _V
43a1i 11 . . . . 5  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  e. 
_V )
5 id 21 . . . . . . . . 9  |-  ( f  =  (Scalar `  w
)  ->  f  =  (Scalar `  w ) )
6 fveq2 5729 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
7 isclm.f . . . . . . . . . 10  |-  F  =  (Scalar `  W )
86, 7syl6eqr 2487 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
95, 8sylan9eqr 2491 . . . . . . . 8  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  f  =  F )
109adantr 453 . . . . . . 7  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
f  =  F )
11 id 21 . . . . . . . . 9  |-  ( k  =  ( Base `  f
)  ->  k  =  ( Base `  f )
)
129fveq2d 5733 . . . . . . . . . 10  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  ( Base `  F
) )
13 isclm.k . . . . . . . . . 10  |-  K  =  ( Base `  F
)
1412, 13syl6eqr 2487 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  K )
1511, 14sylan9eqr 2491 . . . . . . . 8  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
k  =  K )
1615oveq2d 6098 . . . . . . 7  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(flds  k
)  =  (flds  K ) )
1710, 16eqeq12d 2451 . . . . . 6  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( f  =  (flds  k )  <-> 
F  =  (flds  K ) ) )
1815eleq1d 2503 . . . . . 6  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( k  e.  (SubRing ` fld ) 
<->  K  e.  (SubRing ` fld ) ) )
1917, 18anbi12d 693 . . . . 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 3198 . . . 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 3198 . . 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 19089 . . 3  |- CMod  =  {
w  e.  LMod  |  [. (Scalar `  w )  / 
f ]. [. ( Base `  f )  /  k ]. ( f  =  (flds  k )  /\  k  e.  (SubRing ` fld ) ) }
2321, 22elrab2 3095 . 2  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  ( F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) ) )
24 3anass 941 . 2  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  <->  ( W  e.  LMod  /\  ( F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) ) )
2523, 24bitr4i 245 1  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2957   [.wsbc 3162   ` cfv 5455  (class class class)co 6082   Basecbs 13470   ↾s cress 13471  Scalarcsca 13533  SubRingcsubrg 15865   LModclmod 15951  ℂfldccnfld 16704  CModcclm 19088
This theorem is referenced by:  clmsca  19091  clmsubrg  19092  clmlmod  19093  isclmi  19103  lmhmclm  19112  cphclm  19153  tchclm  19190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-nul 4339
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-iota 5419  df-fv 5463  df-ov 6085  df-clm 19089
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