Theorem clmsca 22865

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 e. 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 22864	. 2 |- ( W e. CMod <-> ( W e. LMod /\ F = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) )
4	3	simp2bi 1077	1 |- ( W e. CMod -> F = (ℂfld ↾s K ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858  Scalarcsca 15944  SubRingcsubrg 18776   LModclmod 18863  ℂ_fldccnfld 19746  CModcclm 22862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-clm 22863

This theorem is referenced by:  clm0  22872  clm1  22873  clmadd  22874  clmmul  22875  clmcj  22876  clmsub  22880  clmneg  22881  clmabs  22883  cvsdiv  22932  isncvsngp  22949