This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem reds

Description: The distance of the field of reals. (Contributed by Thierry Arnoux, 20-Jun-2019)

Ref Expression
Assertion reds 
|- ( abs o. - ) = ( dist ` RRfld )

Proof

Step Hyp Ref Expression
1 reex 
 |-  RR e. _V
2 df-refld 
 |-  RRfld = ( CCfld |`s RR )
3 cnfldds 
 |-  ( abs o. - ) = ( dist ` CCfld )
4 2 3 ressds 
 |-  ( RR e. _V -> ( abs o. - ) = ( dist ` RRfld ) )
5 1 4 ax-mp 
 |-  ( abs o. - ) = ( dist ` RRfld )