Theorem cvsclm 25315

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 25313	. . . 4 ⊢ ℂVec = (ℂMod ∩ LVec)
3	2	elin2 4159	. . 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 2146  LVecclvec 21253  ℂModcclm 25251  ℂVecccvs 25312

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 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-in 3915  df-cvs 25313

This theorem is used by:  cvsunit  25320  cvsdiv  25321  cvsmuleqdivd  25323  cvsdiveqd  25324  isncvsngp  25338  ncvsprp  25341  ncvsm1  25343  ncvsdif  25344  ncvspi  25345  ncvspds  25350  cnncvsmulassdemo  25353  ttgcontlem1  29264