Theorem bj-isvec 37005

Description: The predicate "is a vector space". (Contributed by BJ, 6-Jan-2024.)

Hypothesis

Ref	Expression
bj-isvec.scal	⊢ (𝜑 → 𝐾 = (Scalar‘𝑉))

Assertion

Ref	Expression
bj-isvec	⊢ (𝜑 → (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing)))

Proof of Theorem bj-isvec

Step	Hyp	Ref	Expression
1		eqid 2726	. . 3 ⊢ (Scalar‘𝑉) = (Scalar‘𝑉)
2	1	islvec 21076	. 2 ⊢ (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ (Scalar‘𝑉) ∈ DivRing))
3		bj-isvec.scal	. . . . 5 ⊢ (𝜑 → 𝐾 = (Scalar‘𝑉))
4	3	eqcomd 2732	. . . 4 ⊢ (𝜑 → (Scalar‘𝑉) = 𝐾)
5	4	eleq1d 2811	. . 3 ⊢ (𝜑 → ((Scalar‘𝑉) ∈ DivRing ↔ 𝐾 ∈ DivRing))
6	5	anbi2d 628	. 2 ⊢ (𝜑 → ((𝑉 ∈ LMod ∧ (Scalar‘𝑉) ∈ DivRing) ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing)))
7	2, 6	bitrid 282	1 ⊢ (𝜑 → (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ‘cfv 6544  Scalarcsca 17262  DivRingcdr 20701  LModclmod 20830  LVecclvec 21074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3421  df-v 3465  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-br 5145  df-iota 6496  df-fv 6552  df-lvec 21075

This theorem is referenced by:  bj-rvecvec  37017