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 )