Theorem axmulgt0

Description: The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-mulgt0 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005)

	Ref	Expression
Assertion	axmulgt0	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 · 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-pre-mulgt0	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵 ) → 0 <ℝ ( 𝐴 · 𝐵 ) ) )
2		0re	⊢ 0 ∈ ℝ
3		ltxrlt	⊢ ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 < 𝐴 ↔ 0 <ℝ 𝐴 ) )
4	2 3	mpan	⊢ ( 𝐴 ∈ ℝ → ( 0 < 𝐴 ↔ 0 <ℝ 𝐴 ) )
5		ltxrlt	⊢ ( ( 0 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐵 ↔ 0 <ℝ 𝐵 ) )
6	2 5	mpan	⊢ ( 𝐵 ∈ ℝ → ( 0 < 𝐵 ↔ 0 <ℝ 𝐵 ) )
7	4 6	bi2anan9	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) ↔ ( 0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵 ) ) )
8		remulcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 · 𝐵 ) ∈ ℝ )
9		ltxrlt	⊢ ( ( 0 ∈ ℝ ∧ ( 𝐴 · 𝐵 ) ∈ ℝ ) → ( 0 < ( 𝐴 · 𝐵 ) ↔ 0 <ℝ ( 𝐴 · 𝐵 ) ) )
10	2 8 9	sylancr	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < ( 𝐴 · 𝐵 ) ↔ 0 <ℝ ( 𝐴 · 𝐵 ) ) )
11	1 7 10	3imtr4d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 · 𝐵 ) ) )