Theorem xlt0neg1

Description: Extended real version of lt0neg1 . (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	xlt0neg1	⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 < 0 ↔ 0 < -𝑒 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		0xr	⊢ 0 ∈ ℝ*
2		xltneg	⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( 𝐴 < 0 ↔ -𝑒 0 < -𝑒 𝐴 ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 < 0 ↔ -𝑒 0 < -𝑒 𝐴 ) )
4		xneg0	⊢ -𝑒 0 = 0
5	4	breq1i	⊢ ( -𝑒 0 < -𝑒 𝐴 ↔ 0 < -𝑒 𝐴 )
6	3 5	bitrdi	⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 < 0 ↔ 0 < -𝑒 𝐴 ) )