Metamath Proof Explorer

Theorem rrextchr

Description: The ring characteristic of an extension of RR is zero. (Contributed by Thierry Arnoux, 2-May-2018)

Ref Expression
Assertion rrextchr 
|- ( R e. RRExt -> ( chr ` R ) = 0 )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Base ` R ) = ( Base ` R )
2 eqid 
 |-  ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) )
3 eqid 
 |-  ( ZMod ` R ) = ( ZMod ` R )
4 1 2 3 isrrext 
 |-  ( R e. RRExt <-> ( ( R e. NrmRing /\ R e. DivRing ) /\ ( ( ZMod ` R ) e. NrmMod /\ ( chr ` R ) = 0 ) /\ ( R e. CUnifSp /\ ( UnifSt ` R ) = ( metUnif ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) ) ) )
5 4 simp2bi 
 |-  ( R e. RRExt -> ( ( ZMod ` R ) e. NrmMod /\ ( chr ` R ) = 0 ) )
6 5 simprd 
 |-  ( R e. RRExt -> ( chr ` R ) = 0 )