Theorem lvecring 41662

Description: The scalar component of a vector space is a ring. (Contributed by SN, 28-May-2023.)

Hypothesis

Ref	Expression
lvecring.1	⊢ 𝐹 = (Scalar‘𝑊)

Assertion

Ref	Expression
lvecring	⊢ (𝑊 ∈ LVec → 𝐹 ∈ Ring)

Proof of Theorem lvecring

Step	Hyp	Ref	Expression
1		lveclmod 20954	. 2 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
2		lvecring.1	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodring 20714	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
4	1, 3	syl 17	1 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ Ring)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ‘cfv 6537  Scalarcsca 17209  Ringcrg 20138  LModclmod 20706  LVecclvec 20950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-ov 7408  df-lmod 20708  df-lvec 20951

This theorem is referenced by: (None)