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Theorem clmsubrg 18580
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
clmsubrg  |-  ( W  e. CMod  ->  K  e.  (SubRing ` fld ) )

Proof of Theorem clmsubrg
StepHypRef Expression
1 isclm.f . . 3  |-  F  =  (Scalar `  W )
2 isclm.k . . 3  |-  K  =  ( Base `  F
)
31, 2isclm 18578 . 2  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) )
43simp3bi 972 1  |-  ( W  e. CMod  ->  K  e.  (SubRing ` fld ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   ` 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:  clm0  18586  clm1  18587  clmzss  18592  clmsscn  18593  clmsub  18594  clmneg  18595  clmabs  18596  clmacl  18597  clmmcl  18598  clmsubcl  18599
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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