Theorem fldsdrgfldext

Description: A sub-division-ring of a field forms a field extension. (Contributed by Thierry Arnoux, 19-Oct-2025)

	Ref	Expression
Hypotheses	fldsdrgfldext.1	⊢ 𝐺 = ( 𝐹 ↾s 𝐴 )
	fldsdrgfldext.2	⊢ ( 𝜑 → 𝐹 ∈ Field )
	fldsdrgfldext.3	⊢ ( 𝜑 → 𝐴 ∈ ( SubDRing ‘ 𝐹 ) )
Assertion	fldsdrgfldext	⊢ ( 𝜑 → 𝐹 /FldExt 𝐺 )

Proof

Step	Hyp	Ref	Expression
1		fldsdrgfldext.1	⊢ 𝐺 = ( 𝐹 ↾s 𝐴 )
2		fldsdrgfldext.2	⊢ ( 𝜑 → 𝐹 ∈ Field )
3		fldsdrgfldext.3	⊢ ( 𝜑 → 𝐴 ∈ ( SubDRing ‘ 𝐹 ) )
4		fldsdrgfld	⊢ ( ( 𝐹 ∈ Field ∧ 𝐴 ∈ ( SubDRing ‘ 𝐹 ) ) → ( 𝐹 ↾s 𝐴 ) ∈ Field )
5	2 3 4	syl2anc	⊢ ( 𝜑 → ( 𝐹 ↾s 𝐴 ) ∈ Field )
6	1 5	eqeltrid	⊢ ( 𝜑 → 𝐺 ∈ Field )
7		eqid	⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 )
8	7	sdrgss	⊢ ( 𝐴 ∈ ( SubDRing ‘ 𝐹 ) → 𝐴 ⊆ ( Base ‘ 𝐹 ) )
9	1 7	ressbas2	⊢ ( 𝐴 ⊆ ( Base ‘ 𝐹 ) → 𝐴 = ( Base ‘ 𝐺 ) )
10	3 8 9	3syl	⊢ ( 𝜑 → 𝐴 = ( Base ‘ 𝐺 ) )
11	10	oveq2d	⊢ ( 𝜑 → ( 𝐹 ↾s 𝐴 ) = ( 𝐹 ↾s ( Base ‘ 𝐺 ) ) )
12	1 11	eqtrid	⊢ ( 𝜑 → 𝐺 = ( 𝐹 ↾s ( Base ‘ 𝐺 ) ) )
13		sdrgsubrg	⊢ ( 𝐴 ∈ ( SubDRing ‘ 𝐹 ) → 𝐴 ∈ ( SubRing ‘ 𝐹 ) )
14	3 13	syl	⊢ ( 𝜑 → 𝐴 ∈ ( SubRing ‘ 𝐹 ) )
15	10 14	eqeltrrd	⊢ ( 𝜑 → ( Base ‘ 𝐺 ) ∈ ( SubRing ‘ 𝐹 ) )
16		brfldext	⊢ ( ( 𝐹 ∈ Field ∧ 𝐺 ∈ Field ) → ( 𝐹 /FldExt 𝐺 ↔ ( 𝐺 = ( 𝐹 ↾s ( Base ‘ 𝐺 ) ) ∧ ( Base ‘ 𝐺 ) ∈ ( SubRing ‘ 𝐹 ) ) ) )
17	16	biimpar	⊢ ( ( ( 𝐹 ∈ Field ∧ 𝐺 ∈ Field ) ∧ ( 𝐺 = ( 𝐹 ↾s ( Base ‘ 𝐺 ) ) ∧ ( Base ‘ 𝐺 ) ∈ ( SubRing ‘ 𝐹 ) ) ) → 𝐹 /FldExt 𝐺 )
18	2 6 12 15 17	syl22anc	⊢ ( 𝜑 → 𝐹 /FldExt 𝐺 )