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Theorem clmsubrg 19091
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 19089 . 2  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) )
43simp3bi 974 1  |-  ( W  e. CMod  ->  K  e.  (SubRing ` fld ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470  Scalarcsca 13532  SubRingcsubrg 15864   LModclmod 15950  ℂfldccnfld 16703  CModcclm 19087
This theorem is referenced by:  clm0  19097  clm1  19098  clmzss  19103  clmsscn  19104  clmsub  19105  clmneg  19106  clmabs  19107  clmacl  19108  clmmcl  19109  clmsubcl  19110
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-clm 19088
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