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Theorem isclm 20635
Description: A complex module is a left module over a subring of the complex numbers. (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 5700 . . . . 5  |-  (Scalar `  w )  e.  _V
21a1i 11 . . . 4  |-  ( w  =  W  ->  (Scalar `  w )  e.  _V )
3 fvex 5700 . . . . . 6  |-  ( Base `  f )  e.  _V
43a1i 11 . . . . 5  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  e. 
_V )
5 id 22 . . . . . . . . 9  |-  ( f  =  (Scalar `  w
)  ->  f  =  (Scalar `  w ) )
6 fveq2 5690 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
7 isclm.f . . . . . . . . . 10  |-  F  =  (Scalar `  W )
86, 7syl6eqr 2492 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
95, 8sylan9eqr 2496 . . . . . . . 8  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  f  =  F )
109adantr 465 . . . . . . 7  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
f  =  F )
11 id 22 . . . . . . . . 9  |-  ( k  =  ( Base `  f
)  ->  k  =  ( Base `  f )
)
129fveq2d 5694 . . . . . . . . . 10  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  ( Base `  F
) )
13 isclm.k . . . . . . . . . 10  |-  K  =  ( Base `  F
)
1412, 13syl6eqr 2492 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  K )
1511, 14sylan9eqr 2496 . . . . . . . 8  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
k  =  K )
1615oveq2d 6106 . . . . . . 7  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(flds  k
)  =  (flds  K ) )
1710, 16eqeq12d 2456 . . . . . 6  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( f  =  (flds  k )  <-> 
F  =  (flds  K ) ) )
1815eleq1d 2508 . . . . . 6  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( k  e.  (SubRing ` fld ) 
<->  K  e.  (SubRing ` fld ) ) )
1917, 18anbi12d 710 . . . . 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 3222 . . . 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 3222 . . 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 20634 . . 3  |- CMod  =  {
w  e.  LMod  |  [. (Scalar `  w )  / 
f ]. [. ( Base `  f )  /  k ]. ( f  =  (flds  k )  /\  k  e.  (SubRing ` fld ) ) }
2321, 22elrab2 3118 . 2  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  ( F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) ) )
24 3anass 969 . 2  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  <->  ( W  e.  LMod  /\  ( F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) ) )
2523, 24bitr4i 252 1  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2971   [.wsbc 3185   ` cfv 5417  (class class class)co 6090   Basecbs 14173   ↾s cress 14174  Scalarcsca 14240  SubRingcsubrg 16860   LModclmod 16947  ℂfldccnfld 17817  CModcclm 20633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4420
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-iota 5380  df-fv 5425  df-ov 6093  df-clm 20634
This theorem is referenced by:  clmsca  20636  clmsubrg  20637  clmlmod  20638  isclmi  20648  lmhmclm  20657  cphclm  20707  tchclm  20746
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