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Theorem bj-isvec 37289
Description: The predicate "is a vector space". (Contributed by BJ, 6-Jan-2024.)
Hypothesis
Ref Expression
bj-isvec.scal (𝜑𝐾 = (Scalar‘𝑉))
Assertion
Ref Expression
bj-isvec (𝜑 → (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing)))

Proof of Theorem bj-isvec
StepHypRef Expression
1 eqid 2736 . . 3 (Scalar‘𝑉) = (Scalar‘𝑉)
21islvec 21104 . 2 (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ (Scalar‘𝑉) ∈ DivRing))
3 bj-isvec.scal . . . . 5 (𝜑𝐾 = (Scalar‘𝑉))
43eqcomd 2742 . . . 4 (𝜑 → (Scalar‘𝑉) = 𝐾)
54eleq1d 2825 . . 3 (𝜑 → ((Scalar‘𝑉) ∈ DivRing ↔ 𝐾 ∈ DivRing))
65anbi2d 630 . 2 (𝜑 → ((𝑉 ∈ LMod ∧ (Scalar‘𝑉) ∈ DivRing) ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing)))
72, 6bitrid 283 1 (𝜑 → (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  cfv 6560  Scalarcsca 17301  DivRingcdr 20730  LModclmod 20859  LVecclvec 21102
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
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-rab 3436  df-v 3481  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-lvec 21103
This theorem is referenced by:  bj-rvecvec  37301
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