Theorem clmsca 25099

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 = (ℂfld ↾s 𝐾))

Proof of Theorem clmsca

Step	Hyp	Ref	Expression
1		isclm.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
2		isclm.k	. . 3 ⊢ 𝐾 = (Base‘𝐹)
3	1, 2	isclm 25098	. 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
4	3	simp2bi 1146	1 ⊢ (𝑊 ∈ ℂMod → 𝐹 = (ℂfld ↾s 𝐾))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ‘cfv 6560  (class class class)co 7432  Basecbs 17248   ↾_s cress 17275  Scalarcsca 17301  SubRingcsubrg 20570  LModclmod 20859  ℂ_fldccnfld 21365  ℂModcclm 25096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-clm 25097

This theorem is referenced by:  clm0  25106  clm1  25107  clmadd  25108  clmmul  25109  clmcj  25110  clmsub  25114  clmneg  25115  clmabs  25117  cvsdiv  25166  isncvsngp  25184