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Metamath Proof Explorer

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 ) )