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Theorem clmsca 22673
 Description: A complex module is a left module over a subring of the 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
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 22672 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
43simp2bi 1070 1 (𝑊 ∈ ℂMod → 𝐹 = (ℂflds 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696  Scalarcsca 15771  SubRingcsubrg 18599  LModclmod 18686  ℂfldccnfld 19567  ℂModcclm 22670 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-clm 22671 This theorem is referenced by:  clm0  22680  clm1  22681  clmadd  22682  clmmul  22683  clmcj  22684  clmsub  22688  clmneg  22689  clmabs  22691  cvsdiv  22740  isncvsngp  22757
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