Theorem clmsca 24134

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 24133	. 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
4	3	simp2bi 1144	1 ⊢ (𝑊 ∈ ℂMod → 𝐹 = (ℂfld ↾s 𝐾))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ‘cfv 6418  (class class class)co 7255  Basecbs 16840   ↾_s cress 16867  Scalarcsca 16891  SubRingcsubrg 19935  LModclmod 20038  ℂ_fldccnfld 20510  ℂModcclm 24131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-clm 24132

This theorem is referenced by:  clm0  24141  clm1  24142  clmadd  24143  clmmul  24144  clmcj  24145  clmsub  24149  clmneg  24150  clmabs  24152  cvsdiv  24201  isncvsngp  24218