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Theorem clmring 22678
 Description: The scalar ring of a complex 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
StepHypRef Expression
1 clmlmod 22675 . 2 (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
2 clm0.f . . 3 𝐹 = (Scalar‘𝑊)
32lmodring 18694 . 2 (𝑊 ∈ LMod → 𝐹 ∈ Ring)
41, 3syl 17 1 (𝑊 ∈ ℂMod → 𝐹 ∈ Ring)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  Scalarcsca 15771  Ringcrg 18370  LModclmod 18686  ℂModcclm 22670 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-lmod 18688  df-clm 22671 This theorem is referenced by:  clmvsubval  22717  cvsmuleqdivd  22742
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