Theorem fldextchr

Description: The characteristic of a subfield is the same as the characteristic of the larger field. (Contributed by Thierry Arnoux, 20-Aug-2023)

	Ref	Expression
Assertion	fldextchr	⊢ ( 𝐸 /FldExt 𝐹 → ( chr ‘ 𝐹 ) = ( chr ‘ 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		fldextress	⊢ ( 𝐸 /FldExt 𝐹 → 𝐹 = ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) )
2	1	fveq2d	⊢ ( 𝐸 /FldExt 𝐹 → ( chr ‘ 𝐹 ) = ( chr ‘ ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) ) )
3		eqid	⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 )
4	3	fldextsubrg	⊢ ( 𝐸 /FldExt 𝐹 → ( Base ‘ 𝐹 ) ∈ ( SubRing ‘ 𝐸 ) )
5		subrgchr	⊢ ( ( Base ‘ 𝐹 ) ∈ ( SubRing ‘ 𝐸 ) → ( chr ‘ ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) ) = ( chr ‘ 𝐸 ) )
6	4 5	syl	⊢ ( 𝐸 /FldExt 𝐹 → ( chr ‘ ( 𝐸 ↾s ( Base ‘ 𝐹 ) ) ) = ( chr ‘ 𝐸 ) )
7	2 6	eqtrd	⊢ ( 𝐸 /FldExt 𝐹 → ( chr ‘ 𝐹 ) = ( chr ‘ 𝐸 ) )