Theorem reds

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

	Ref	Expression
Assertion	reds	⊢ ( abs ∘ − ) = ( dist ‘ ℝfld )

Proof

Step	Hyp	Ref	Expression
1		reex	⊢ ℝ ∈ V
2		df-refld	⊢ ℝfld = ( ℂfld ↾s ℝ )
3		cnfldds	⊢ ( abs ∘ − ) = ( dist ‘ ℂfld )
4	2 3	ressds	⊢ ( ℝ ∈ V → ( abs ∘ − ) = ( dist ‘ ℝfld ) )
5	1 4	ax-mp	⊢ ( abs ∘ − ) = ( dist ‘ ℝfld )