This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )