Theorem elfzolem1

Description: A member in a half-open integer interval is less than or equal to the upper bound minus 1 . (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Assertion	elfzolem1	|- ( K e. ( M ..^ N ) -> K <_ ( N - 1 ) )

Proof

Step	Hyp	Ref	Expression
1		id	|- ( K e. ( M ..^ N ) -> K e. ( M ..^ N ) )
2		elfzoel2	|- ( K e. ( M ..^ N ) -> N e. ZZ )
3		simpl	|- ( ( K e. ( M ..^ N ) /\ N e. ZZ ) -> K e. ( M ..^ N ) )
4		fzoval	|- ( N e. ZZ -> ( M ..^ N ) = ( M ... ( N - 1 ) ) )
5	4	adantl	|- ( ( K e. ( M ..^ N ) /\ N e. ZZ ) -> ( M ..^ N ) = ( M ... ( N - 1 ) ) )
6	3 5	eleqtrd	|- ( ( K e. ( M ..^ N ) /\ N e. ZZ ) -> K e. ( M ... ( N - 1 ) ) )
7		elfzle2	|- ( K e. ( M ... ( N - 1 ) ) -> K <_ ( N - 1 ) )
8	6 7	syl	|- ( ( K e. ( M ..^ N ) /\ N e. ZZ ) -> K <_ ( N - 1 ) )
9	1 2 8	syl2anc	|- ( K e. ( M ..^ N ) -> K <_ ( N - 1 ) )