Theorem flbi

Description: A condition equivalent to floor. (Contributed by NM, 11-Mar-2005) (Revised by Mario Carneiro, 2-Nov-2013)

	Ref	Expression
Assertion	flbi	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( ( ⌊ ‘ 𝐴 ) = 𝐵 ↔ ( 𝐵 ≤ 𝐴 ∧ 𝐴 < ( 𝐵 + 1 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		flval	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) = ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) )
2	1	eqeq1d	⊢ ( 𝐴 ∈ ℝ → ( ( ⌊ ‘ 𝐴 ) = 𝐵 ↔ ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) = 𝐵 ) )
3	2	adantr	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( ( ⌊ ‘ 𝐴 ) = 𝐵 ↔ ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) = 𝐵 ) )
4		rebtwnz	⊢ ( 𝐴 ∈ ℝ → ∃! 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) )
5		breq1	⊢ ( 𝑥 = 𝐵 → ( 𝑥 ≤ 𝐴 ↔ 𝐵 ≤ 𝐴 ) )
6		oveq1	⊢ ( 𝑥 = 𝐵 → ( 𝑥 + 1 ) = ( 𝐵 + 1 ) )
7	6	breq2d	⊢ ( 𝑥 = 𝐵 → ( 𝐴 < ( 𝑥 + 1 ) ↔ 𝐴 < ( 𝐵 + 1 ) ) )
8	5 7	anbi12d	⊢ ( 𝑥 = 𝐵 → ( ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ↔ ( 𝐵 ≤ 𝐴 ∧ 𝐴 < ( 𝐵 + 1 ) ) ) )
9	8	riota2	⊢ ( ( 𝐵 ∈ ℤ ∧ ∃! 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) → ( ( 𝐵 ≤ 𝐴 ∧ 𝐴 < ( 𝐵 + 1 ) ) ↔ ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) = 𝐵 ) )
10	4 9	sylan2	⊢ ( ( 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℝ ) → ( ( 𝐵 ≤ 𝐴 ∧ 𝐴 < ( 𝐵 + 1 ) ) ↔ ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) = 𝐵 ) )
11	10	ancoms	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐵 ≤ 𝐴 ∧ 𝐴 < ( 𝐵 + 1 ) ) ↔ ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) = 𝐵 ) )
12	3 11	bitr4d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( ( ⌊ ‘ 𝐴 ) = 𝐵 ↔ ( 𝐵 ≤ 𝐴 ∧ 𝐴 < ( 𝐵 + 1 ) ) ) )