Theorem fldextress

Description: Field extension implies a structure restriction relation. (Contributed by Thierry Arnoux, 29-Jul-2023)

	Ref	Expression
Assertion	fldextress	⊢ ( 𝐸 /FldExt 𝐹 → 𝐹 = ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fldextfld1	⊢ ( 𝐸 /FldExt 𝐹 → 𝐸 ∈ Field )
2		fldextfld2	⊢ ( 𝐸 /FldExt 𝐹 → 𝐹 ∈ Field )
3		brfldext	⊢ ( ( 𝐸 ∈ Field ∧ 𝐹 ∈ Field ) → ( 𝐸 /FldExt 𝐹 ↔ ( 𝐹 = ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) ∧ ( Base ‘ 𝐹 ) ∈ ( SubRing ‘ 𝐸 ) ) ) )
4	1 2 3	syl2anc	⊢ ( 𝐸 /FldExt 𝐹 → ( 𝐸 /FldExt 𝐹 ↔ ( 𝐹 = ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) ∧ ( Base ‘ 𝐹 ) ∈ ( SubRing ‘ 𝐸 ) ) ) )
5	4	ibi	⊢ ( 𝐸 /FldExt 𝐹 → ( 𝐹 = ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) ∧ ( Base ‘ 𝐹 ) ∈ ( SubRing ‘ 𝐸 ) ) )
6	5	simpld	⊢ ( 𝐸 /FldExt 𝐹 → 𝐹 = ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) )