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Metamath Proof Explorer

Theorem ply1lvec

Description: In a division ring, the univariate polynomials form a vector space. (Contributed by Thierry Arnoux, 19-Feb-2025)

Ref Expression
Hypotheses ply1lvec.p 
|- P = ( Poly1 ` R )
ply1lvec.r 
|- ( ph -> R e. DivRing )
Assertion ply1lvec 
|- ( ph -> P e. LVec )

Proof

Step Hyp Ref Expression
1 ply1lvec.p 
 |-  P = ( Poly1 ` R )
2 ply1lvec.r 
 |-  ( ph -> R e. DivRing )
3 2 drngringd 
 |-  ( ph -> R e. Ring )
4 1 ply1lmod 
 |-  ( R e. Ring -> P e. LMod )
5 3 4 syl 
 |-  ( ph -> P e. LMod )
6 1 ply1sca 
 |-  ( R e. DivRing -> R = ( Scalar ` P ) )
7 2 6 syl 
 |-  ( ph -> R = ( Scalar ` P ) )
8 7 2 eqeltrrd 
 |-  ( ph -> ( Scalar ` P ) e. DivRing )
9 eqid 
 |-  ( Scalar ` P ) = ( Scalar ` P )
10 9 islvec 
 |-  ( P e. LVec <-> ( P e. LMod /\ ( Scalar ` P ) e. DivRing ) )
11 5 8 10 sylanbrc 
 |-  ( ph -> P e. LVec )