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Metamath Proof Explorer

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 ∈ ( 𝑀 ..^ 𝑁 ) → 𝐾 ≤ ( 𝑁 − 1 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) )
2		elfzoel2	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑁 ∈ ℤ )
3		simpl	⊢ ( ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝑁 ∈ ℤ ) → 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) )
4		fzoval	⊢ ( 𝑁 ∈ ℤ → ( 𝑀 ..^ 𝑁 ) = ( 𝑀 ... ( 𝑁 − 1 ) ) )
5	4	adantl	⊢ ( ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ..^ 𝑁 ) = ( 𝑀 ... ( 𝑁 − 1 ) ) )
6	3 5	eleqtrd	⊢ ( ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝑁 ∈ ℤ ) → 𝐾 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) )
7		elfzle2	⊢ ( 𝐾 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) → 𝐾 ≤ ( 𝑁 − 1 ) )
8	6 7	syl	⊢ ( ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝑁 ∈ ℤ ) → 𝐾 ≤ ( 𝑁 − 1 ) )
9	1 2 8	syl2anc	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝐾 ≤ ( 𝑁 − 1 ) )