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Theorem clmsubrg 23674
 Description: The base set of the ring of scalars of a subcomplex module is the base set of a subring of the field of 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
clmsubrg (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld))

Proof of Theorem clmsubrg
StepHypRef Expression
1 isclm.f . . 3 𝐹 = (Scalar‘𝑊)
2 isclm.k . . 3 𝐾 = (Base‘𝐹)
31, 2isclm 23672 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
43simp3bi 1144 1 (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  ‘cfv 6343  (class class class)co 7149  Basecbs 16483   ↾s cress 16484  Scalarcsca 16568  SubRingcsubrg 19531  LModclmod 19634  ℂfldccnfld 20545  ℂModcclm 23670 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152  df-clm 23671 This theorem is referenced by:  clm0  23680  clm1  23681  clmzss  23686  clmsscn  23687  clmsub  23688  clmneg  23689  clmabs  23691  clmacl  23692  clmmcl  23693  clmsubcl  23694  cmodscexp  23729  cvsdiv  23740  isncvsngp  23757
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