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Theorem clmsca 23648
Description: The ring of scalars 𝐹 of a subcomplex module is the restriction of the field of complex numbers to the base set of 𝐹. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
isclm.f 𝐹 = (Scalar‘𝑊)
isclm.k 𝐾 = (Base‘𝐹)
Assertion
Ref Expression
clmsca (𝑊 ∈ ℂMod → 𝐹 = (ℂflds 𝐾))

Proof of Theorem clmsca
StepHypRef Expression
1 isclm.f . . 3 𝐹 = (Scalar‘𝑊)
2 isclm.k . . 3 𝐾 = (Base‘𝐹)
31, 2isclm 23647 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
43simp2bi 1143 1 (𝑊 ∈ ℂMod → 𝐹 = (ℂflds 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  cfv 6328  (class class class)co 7130  Basecbs 16461  s cress 16462  Scalarcsca 16546  SubRingcsubrg 19506  LModclmod 19609  fldccnfld 20520  ℂModcclm 23645
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 2178  ax-ext 2793  ax-nul 5183
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 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336  df-ov 7133  df-clm 23646
This theorem is referenced by:  clm0  23655  clm1  23656  clmadd  23657  clmmul  23658  clmcj  23659  clmsub  23663  clmneg  23664  clmabs  23666  cvsdiv  23715  isncvsngp  23732
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