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Theorem isclm 18562
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 5539 . . . . 5  |-  (Scalar `  w )  e.  _V
21a1i 10 . . . 4  |-  ( w  =  W  ->  (Scalar `  w )  e.  _V )
3 fvex 5539 . . . . . 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 5525 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
7 isclm.f . . . . . . . . . 10  |-  F  =  (Scalar `  W )
86, 7syl6eqr 2333 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
95, 8sylan9eqr 2337 . . . . . . . 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 5529 . . . . . . . . . 10  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  ( Base `  F
) )
13 isclm.k . . . . . . . . . 10  |-  K  =  ( Base `  F
)
1412, 13syl6eqr 2333 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  K )
1511, 14sylan9eqr 2337 . . . . . . . 8  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
k  =  K )
1615oveq2d 5874 . . . . . . 7  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(flds  k
)  =  (flds  K ) )
1710, 16eqeq12d 2297 . . . . . 6  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( f  =  (flds  k )  <-> 
F  =  (flds  K ) ) )
1815eleq1d 2349 . . . . . 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 3027 . . . 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 3027 . . 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 18561 . . 3  |- CMod  =  {
w  e.  LMod  |  [. (Scalar `  w )  / 
f ]. [. ( Base `  f )  /  k ]. ( f  =  (flds  k )  /\  k  e.  (SubRing ` fld ) ) }
2321, 22elrab2 2925 . 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 1623    e. wcel 1684   _Vcvv 2788   [.wsbc 2991   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149  Scalarcsca 13211  SubRingcsubrg 15541   LModclmod 15627  ℂfldccnfld 16377  CModcclm 18560
This theorem is referenced by:  clmsca  18563  clmsubrg  18564  clmlmod  18565  isclmi  18575  lmhmclm  18584  cphclm  18625  tchclm  18662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-clm 18561
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