Theorem clmsubrg 25396

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 25394	. 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
4	3	simp3bi 1165	1 ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ‘cfv 6543  (class class class)co 7424  Basecbs 17396   ↾_s cress 17417  Scalarcsca 17440  SubRingcsubrg 20830  LModclmod 21144  ℂ_fldccnfld 21687  ℂModcclm 25392

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6497  df-fv 6551  df-ov 7427  df-clm 25393

This theorem is used by:  clm0  25402  clm1  25403  clmzss  25408  clmsscn  25409  clmsub  25410  clmneg  25411  clmabs  25413  clmacl  25414  clmmcl  25415  clmsubcl  25416  cmodscexp  25451  cvsdiv  25462  isncvsngp  25479