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Theorem clmsubrg 18564
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 18562 . 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 1623    e. wcel 1684   ` 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:  clm0  18570  clm1  18571  clmzss  18576  clmsscn  18577  clmsub  18578  clmneg  18579  clmabs  18580  clmacl  18581  clmmcl  18582  clmsubcl  18583
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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