Theorem bj-isvec 34572

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 2821	. . 3 ⊢ (Scalar‘𝑉) = (Scalar‘𝑉)
2	1	islvec 19876	. 2 ⊢ (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ (Scalar‘𝑉) ∈ DivRing))
3		bj-isvec.scal	. . . . 5 ⊢ (𝜑 → 𝐾 = (Scalar‘𝑉))
4	3	eqcomd 2827	. . . 4 ⊢ (𝜑 → (Scalar‘𝑉) = 𝐾)
5	4	eleq1d 2897	. . 3 ⊢ (𝜑 → ((Scalar‘𝑉) ∈ DivRing ↔ 𝐾 ∈ DivRing))
6	5	anbi2d 630	. 2 ⊢ (𝜑 → ((𝑉 ∈ LMod ∧ (Scalar‘𝑉) ∈ DivRing) ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing)))
7	2, 6	syl5bb 285	1 ⊢ (𝜑 → (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ‘cfv 6355  Scalarcsca 16568  DivRingcdr 19502  LModclmod 19634  LVecclvec 19874

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-iota 6314  df-fv 6363  df-lvec 19875

This theorem is referenced by:  bj-rvecvec  34583