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Metamath Proof Explorer

Theorem elfzouz2

Description: The upper bound of a half-open range is greater than or equal to an element of the range. (Contributed by Mario Carneiro, 29-Sep-2015)

Ref Expression
Assertion elfzouz2 ⊢ K ∈ M ..^ N → N ∈ ℤ ≥ K

Proof

Step Hyp Ref Expression
1 elfzoelz ⊢ K ∈ M ..^ N → K ∈ ℤ
2 elfzoel2 ⊢ K ∈ M ..^ N → N ∈ ℤ
3 elfzolt2 ⊢ K ∈ M ..^ N → K < N
4 zre ⊢ K ∈ ℤ → K ∈ ℝ
5 zre ⊢ N ∈ ℤ → N ∈ ℝ
6 ltle ⊢ K ∈ ℝ ∧ N ∈ ℝ → K < N → K ≤ N
7 4 5 6 syl2an ⊢ K ∈ ℤ ∧ N ∈ ℤ → K < N → K ≤ N
8 1 2 7 syl2anc ⊢ K ∈ M ..^ N → K < N → K ≤ N
9 3 8 mpd ⊢ K ∈ M ..^ N → K ≤ N
10 eluz2 ⊢ N ∈ ℤ ≥ K ↔ K ∈ ℤ ∧ N ∈ ℤ ∧ K ≤ N
11 1 2 9 10 syl3anbrc ⊢ K ∈ M ..^ N → N ∈ ℤ ≥ K