Theorem xleneg

Description: Extended real version of leneg . (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	xleneg	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ↔ -𝑒 𝐵 ≤ -𝑒 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		xltneg	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐵 < 𝐴 ↔ -𝑒 𝐴 < -𝑒 𝐵 ) )
2	1	ancoms	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐵 < 𝐴 ↔ -𝑒 𝐴 < -𝑒 𝐵 ) )
3	2	notbid	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ¬ 𝐵 < 𝐴 ↔ ¬ -𝑒 𝐴 < -𝑒 𝐵 ) )
4		xrlenlt	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
5		xnegcl	⊢ ( 𝐵 ∈ ℝ* → -𝑒 𝐵 ∈ ℝ* )
6		xnegcl	⊢ ( 𝐴 ∈ ℝ* → -𝑒 𝐴 ∈ ℝ* )
7		xrlenlt	⊢ ( ( -𝑒 𝐵 ∈ ℝ* ∧ -𝑒 𝐴 ∈ ℝ* ) → ( -𝑒 𝐵 ≤ -𝑒 𝐴 ↔ ¬ -𝑒 𝐴 < -𝑒 𝐵 ) )
8	5 6 7	syl2anr	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( -𝑒 𝐵 ≤ -𝑒 𝐴 ↔ ¬ -𝑒 𝐴 < -𝑒 𝐵 ) )
9	3 4 8	3bitr4d	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ↔ -𝑒 𝐵 ≤ -𝑒 𝐴 ) )