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Metamath Proof Explorer

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 ) ) )