Theorem clmsca 25057

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 25056	. 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
4	3	simp2bi 1152	1 ⊢ (𝑊 ∈ ℂMod → 𝐹 = (ℂfld ↾s 𝐾))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ‘cfv 6492  (class class class)co 7363  Basecbs 17177   ↾_s cress 17198  Scalarcsca 17221  SubRingcsubrg 20548  LModclmod 20857  ℂ_fldccnfld 21354  ℂModcclm 25054

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 2712  ax-nul 5235

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 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-clm 25055

This theorem is referenced by:  clm0  25064  clm1  25065  clmadd  25066  clmmul  25067  clmcj  25068  clmsub  25072  clmneg  25073  clmabs  25075  cvsdiv  25124  isncvsngp  25141