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 ) e. ( SubRing ` E ) )
5		subrgchr	|- ( ( Base ` F ) e. ( 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 ) )