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Metamath Proof Explorer

Theorem flltnz

Description: The floor of a non-integer real is less than it. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Assertion flltnz ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) → ( ⌊ ‘ 𝐴 ) < 𝐴 )

Proof

Step Hyp Ref Expression
1 reflcl ⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℝ )
2 1 adantr ⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) → ( ⌊ ‘ 𝐴 ) ∈ ℝ )
3 simpl ⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) → 𝐴 ∈ ℝ )
4 fllelt ⊢ ( 𝐴 ∈ ℝ → ( ( ⌊ ‘ 𝐴 ) ≤ 𝐴 ∧ 𝐴 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) )
5 4 adantr ⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) → ( ( ⌊ ‘ 𝐴 ) ≤ 𝐴 ∧ 𝐴 < ( ( ⌊ ‘ 𝐴 ) + 1 ) ) )
6 5 simpld ⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) → ( ⌊ ‘ 𝐴 ) ≤ 𝐴 )
7 simpr ⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) → ¬ 𝐴 ∈ ℤ )
8 flidz ⊢ ( 𝐴 ∈ ℝ → ( ( ⌊ ‘ 𝐴 ) = 𝐴 ↔ 𝐴 ∈ ℤ ) )
9 8 adantr ⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) → ( ( ⌊ ‘ 𝐴 ) = 𝐴 ↔ 𝐴 ∈ ℤ ) )
10 7 9 mtbird ⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) → ¬ ( ⌊ ‘ 𝐴 ) = 𝐴 )
11 10 neqned ⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) → ( ⌊ ‘ 𝐴 ) ≠ 𝐴 )
12 11 necomd ⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) → 𝐴 ≠ ( ⌊ ‘ 𝐴 ) )
13 2 3 6 12 leneltd ⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) → ( ⌊ ‘ 𝐴 ) < 𝐴 )