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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 = (Scalarβ€˜π‘Š)
Assertion
Ref Expression
lvecring (π‘Š ∈ LVec β†’ 𝐹 ∈ Ring)

Proof of Theorem lvecring
StepHypRef Expression
1 lveclmod 20954 . 2 (π‘Š ∈ LVec β†’ π‘Š ∈ LMod)
2 lvecring.1 . . 3 𝐹 = (Scalarβ€˜π‘Š)
32lmodring 20714 . 2 (π‘Š ∈ LMod β†’ 𝐹 ∈ Ring)
41, 3syl 17 1 (π‘Š ∈ LVec β†’ 𝐹 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜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)
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