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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 Scalar
isclm.k
Assertion
Ref Expression
clmsubrg CMod SubRingfld

Proof of Theorem clmsubrg
StepHypRef Expression
1 isclm.f . . 3 Scalar
2 isclm.k . . 3
31, 2isclm 19089 . 2 CMod flds SubRingfld
43simp3bi 974 1 CMod SubRingfld
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  cfv 5454  (class class class)co 6081  cbs 13469   ↾s cress 13470  Scalarcsca 13532  SubRingcsubrg 15864  clmod 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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