Theorem bj-vecssmod 36683

Description: Vector spaces are modules. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-vecssmod	⊢ LVec ⊆ LMod

Proof of Theorem bj-vecssmod

Step	Hyp	Ref	Expression
1		df-lvec 20970	. 2 ⊢ LVec = {𝑥 ∈ LMod ∣ (Scalar‘𝑥) ∈ DivRing}
2		ssrab2 4073	. 2 ⊢ {𝑥 ∈ LMod ∣ (Scalar‘𝑥) ∈ DivRing} ⊆ LMod
3	1, 2	eqsstri 4012	1 ⊢ LVec ⊆ LMod

Syntax hints:   ∈ wcel 2099  {crab 3427   ⊆ wss 3944  ‘cfv 6542  Scalarcsca 17221  DivRingcdr 20606  LModclmod 20725  LVecclvec 20969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-in 3951  df-ss 3961  df-lvec 20970

This theorem is referenced by:  bj-vecssmodel  36684