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Theorem clmsca 24238
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 24237 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
43simp2bi 1145 1 (𝑊 ∈ ℂMod → 𝐹 = (ℂflds 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  cfv 6426  (class class class)co 7267  Basecbs 16922  s cress 16951  Scalarcsca 16975  SubRingcsubrg 20030  LModclmod 20133  fldccnfld 20607  ℂModcclm 24235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5228
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-iota 6384  df-fv 6434  df-ov 7270  df-clm 24236
This theorem is referenced by:  clm0  24245  clm1  24246  clmadd  24247  clmmul  24248  clmcj  24249  clmsub  24253  clmneg  24254  clmabs  24256  cvsdiv  24305  isncvsngp  24323
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