Theorem cvsclm 25360

Description: A subcomplex vector space is a subcomplex module. (Contributed by Thierry Arnoux, 22-May-2019.)

Hypothesis

Ref	Expression
cvslvec.1	⊢ (𝜑 → 𝑊 ∈ ℂVec)

Assertion

Ref	Expression
cvsclm	⊢ (𝜑 → 𝑊 ∈ ℂMod)

Proof of Theorem cvsclm

Step	Hyp	Ref	Expression
1		cvslvec.1	. 2 ⊢ (𝜑 → 𝑊 ∈ ℂVec)
2		df-cvs 25358	. . . 4 ⊢ ℂVec = (ℂMod ∩ LVec)
3	2	elin2 4152	. . 3 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
4	3	simplbi 502	. 2 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod)
5	1, 4	syl 18	1 ⊢ (𝜑 → 𝑊 ∈ ℂMod)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  LVecclvec 21292  ℂModcclm 25296  ℂVecccvs 25357

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-in 3909  df-cvs 25358

This theorem is used by:  cvsunit  25365  cvsdiv  25366  cvsmuleqdivd  25368  cvsdiveqd  25369  isncvsngp  25383  ncvsprp  25386  ncvsm1  25388  ncvsdif  25389  ncvspi  25390  ncvspds  25395  cnncvsmulassdemo  25398  ttgcontlem1  29349