MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isclm Structured version   Unicode version

Theorem isclm 21327
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 5876 . . . . 5  |-  (Scalar `  w )  e.  _V
21a1i 11 . . . 4  |-  ( w  =  W  ->  (Scalar `  w )  e.  _V )
3 fvex 5876 . . . . . 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 5866 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
7 isclm.f . . . . . . . . . 10  |-  F  =  (Scalar `  W )
86, 7syl6eqr 2526 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
95, 8sylan9eqr 2530 . . . . . . . 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 5870 . . . . . . . . . 10  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  ( Base `  F
) )
13 isclm.k . . . . . . . . . 10  |-  K  =  ( Base `  F
)
1412, 13syl6eqr 2526 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  K )
1511, 14sylan9eqr 2530 . . . . . . . 8  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
k  =  K )
1615oveq2d 6300 . . . . . . 7  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(flds  k
)  =  (flds  K ) )
1710, 16eqeq12d 2489 . . . . . 6  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( f  =  (flds  k )  <-> 
F  =  (flds  K ) ) )
1815eleq1d 2536 . . . . . 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 3368 . . . 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 3368 . . 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 21326 . . 3  |- CMod  =  {
w  e.  LMod  |  [. (Scalar `  w )  / 
f ]. [. ( Base `  f )  /  k ]. ( f  =  (flds  k )  /\  k  e.  (SubRing ` fld ) ) }
2321, 22elrab2 3263 . 2  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  ( F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) ) )
24 3anass 977 . 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 973    = wceq 1379    e. wcel 1767   _Vcvv 3113   [.wsbc 3331   ` cfv 5588  (class class class)co 6284   Basecbs 14490   ↾s cress 14491  Scalarcsca 14558  SubRingcsubrg 17225   LModclmod 17312  ℂfldccnfld 18219  CModcclm 21325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6287  df-clm 21326
This theorem is referenced by:  clmsca  21328  clmsubrg  21329  clmlmod  21330  isclmi  21340  lmhmclm  21349  cphclm  21399  tchclm  21438
  Copyright terms: Public domain W3C validator