Theorem clmring 25104

Description: The scalar ring of a subcomplex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)

Hypothesis

Ref	Expression
clm0.f	⊢ 𝐹 = (Scalar‘𝑊)

Assertion

Ref	Expression
clmring	⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Ring)

Proof of Theorem clmring

Step	Hyp	Ref	Expression
1		clmlmod 25101	. 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
2		clm0.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodring 20867	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
4	1, 3	syl 17	1 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Ring)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ‘cfv 6560  Scalarcsca 17301  Ringcrg 20231  LModclmod 20859  ℂModcclm 25096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-lmod 20861  df-clm 25097

This theorem is referenced by:  clmvsubval  25143  cvsmuleqdivd  25168