Metamath Proof Explorer

Theorem fz1fzo0m1

Description: Translation of one between closed and open integer ranges. (Contributed by Thierry Arnoux, 28-Jul-2020)

		Ref	Expression
	Assertion	fz1fzo0m1	$⊢ M \in (1 \dots N) \to M - 1 \in (0 ..^N)$

Step	Hyp	Ref	Expression
1		elfzmlbm	$⊢ M \in (1 \dots N) \to M - 1 \in (0 \dots N - 1)$
2		elfzel2	$⊢ M \in (1 \dots N) \to N \in ℤ$
3		fzoval	$⊢ N \in ℤ \to (0 ..^N) = (0 \dots N - 1)$
4	2 3	syl	$⊢ M \in (1 \dots N) \to (0 ..^N) = (0 \dots N - 1)$
5	1 4	eleqtrrd	$⊢ M \in (1 \dots N) \to M - 1 \in (0 ..^N)$