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Metamath Proof Explorer

Theorem fldextsralvec

Description: The subring algebra associated with a field extension is a vector space. (Contributed by Thierry Arnoux, 3-Aug-2023)

Ref Expression
Assertion fldextsralvec ( 𝐸 /FldExt 𝐹 → ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) ∈ LVec )

Proof

Step Hyp Ref Expression
1 fldextfld1 ⊢ ( 𝐸 /FldExt 𝐹 → 𝐸 ∈ Field )
2 isfld ⊢ ( 𝐸 ∈ Field ↔ ( 𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing ) )
3 1 2 sylib ⊢ ( 𝐸 /FldExt 𝐹 → ( 𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing ) )
4 3 simpld ⊢ ( 𝐸 /FldExt 𝐹 → 𝐸 ∈ DivRing )
5 fldextress ⊢ ( 𝐸 /FldExt 𝐹 → 𝐹 = ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) )
6 fldextfld2 ⊢ ( 𝐸 /FldExt 𝐹 → 𝐹 ∈ Field )
7 isfld ⊢ ( 𝐹 ∈ Field ↔ ( 𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing ) )
8 6 7 sylib ⊢ ( 𝐸 /FldExt 𝐹 → ( 𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing ) )
9 8 simpld ⊢ ( 𝐸 /FldExt 𝐹 → 𝐹 ∈ DivRing )
10 5 9 eqeltrrd ⊢ ( 𝐸 /FldExt 𝐹 → ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) ∈ DivRing )
11 eqid ⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 )
12 11 fldextsubrg ⊢ ( 𝐸 /FldExt 𝐹 → ( Base ‘ 𝐹 ) ∈ ( SubRing ‘ 𝐸 ) )
13 eqid ⊢ ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) = ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) )
14 eqid ⊢ ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) = ( 𝐸 ↾s ( Base ‘ 𝐹 ) )
15 13 14 sralvec ⊢ ( ( 𝐸 ∈ DivRing ∧ ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) ∈ DivRing ∧ ( Base ‘ 𝐹 ) ∈ ( SubRing ‘ 𝐸 ) ) → ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) ∈ LVec )
16 4 10 12 15 syl3anc ⊢ ( 𝐸 /FldExt 𝐹 → ( ( subringAlg ‘ 𝐸 ) ‘ ( Base ‘ 𝐹 ) ) ∈ LVec )