Theorem clmring 25000

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 24997	. 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
2		clm0.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodring 20805	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
4	1, 3	syl 17	1 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Ring)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ‘cfv 6488  Scalarcsca 17168  Ringcrg 20155  LModclmod 20797  ℂModcclm 24992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-nul 5248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6444  df-fv 6496  df-ov 7357  df-lmod 20799  df-clm 24993

This theorem is referenced by:  clmvsubval  25039  cvsmuleqdivd  25064