Theorem ehl1eudisval

Description: The value of the Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023)

	Ref	Expression
Hypotheses	ehl1eudis.e	|- E = ( EEhil ` 1 )
	ehl1eudis.x	|- X = ( RR ^m { 1 } )
	ehl1eudis.d	|- D = ( dist ` E )
Assertion	ehl1eudisval	|- ( ( F e. X /\ G e. X ) -> ( F D G ) = ( abs ` ( ( F ` 1 ) - ( G ` 1 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ehl1eudis.e	|- E = ( EEhil ` 1 )
2		ehl1eudis.x	|- X = ( RR ^m { 1 } )
3		ehl1eudis.d	|- D = ( dist ` E )
4		fveq1	|- ( x = F -> ( x ` 1 ) = ( F ` 1 ) )
5	4	fvoveq1d	|- ( x = F -> ( abs ` ( ( x ` 1 ) - ( y ` 1 ) ) ) = ( abs ` ( ( F ` 1 ) - ( y ` 1 ) ) ) )
6		fveq1	|- ( y = G -> ( y ` 1 ) = ( G ` 1 ) )
7	6	oveq2d	|- ( y = G -> ( ( F ` 1 ) - ( y ` 1 ) ) = ( ( F ` 1 ) - ( G ` 1 ) ) )
8	7	fveq2d	|- ( y = G -> ( abs ` ( ( F ` 1 ) - ( y ` 1 ) ) ) = ( abs ` ( ( F ` 1 ) - ( G ` 1 ) ) ) )
9	1 2 3	ehl1eudis	|- D = ( x e. X , y e. X |-> ( abs ` ( ( x ` 1 ) - ( y ` 1 ) ) ) )
10		fvex	|- ( abs ` ( ( F ` 1 ) - ( G ` 1 ) ) ) e. _V
11	5 8 9 10	ovmpo	|- ( ( F e. X /\ G e. X ) -> ( F D G ) = ( abs ` ( ( F ` 1 ) - ( G ` 1 ) ) ) )