Theorem flwordi

Description: Ordering relation for the floor function. (Contributed by NM, 31-Dec-2005) (Proof shortened by Fan Zheng, 14-Jul-2016)

	Ref	Expression
Assertion	flwordi	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( ⌊ ‘ 𝐴 ) ≤ ( ⌊ ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		simp1	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ℝ )
2	1	flcld	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( ⌊ ‘ 𝐴 ) ∈ ℤ )
3	2	zred	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( ⌊ ‘ 𝐴 ) ∈ ℝ )
4		simp2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ℝ )
5		flle	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ≤ 𝐴 )
6	1 5	syl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( ⌊ ‘ 𝐴 ) ≤ 𝐴 )
7		simp3	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ≤ 𝐵 )
8	3 1 4 6 7	letrd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( ⌊ ‘ 𝐴 ) ≤ 𝐵 )
9		flge	⊢ ( ( 𝐵 ∈ ℝ ∧ ( ⌊ ‘ 𝐴 ) ∈ ℤ ) → ( ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ↔ ( ⌊ ‘ 𝐴 ) ≤ ( ⌊ ‘ 𝐵 ) ) )
10	4 2 9	syl2anc	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( ( ⌊ ‘ 𝐴 ) ≤ 𝐵 ↔ ( ⌊ ‘ 𝐴 ) ≤ ( ⌊ ‘ 𝐵 ) ) )
11	8 10	mpbid	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( ⌊ ‘ 𝐴 ) ≤ ( ⌊ ‘ 𝐵 ) )