Theorem clmsubrg 25051

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

Step	Hyp	Ref	Expression
1		isclm.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
2		isclm.k	. . 3 ⊢ 𝐾 = (Base‘𝐹)
3	1, 2	isclm 25049	. 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
4	3	simp3bi 1153	1 ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ‘cfv 6485  (class class class)co 7356  Basecbs 17170   ↾_s cress 17191  Scalarcsca 17214  SubRingcsubrg 20541  LModclmod 20850  ℂ_fldccnfld 21347  ℂModcclm 25047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359  df-clm 25048

This theorem is referenced by:  clm0  25057  clm1  25058  clmzss  25063  clmsscn  25064  clmsub  25065  clmneg  25066  clmabs  25068  clmacl  25069  clmmcl  25070  clmsubcl  25071  cmodscexp  25106  cvsdiv  25117  isncvsngp  25134