mul_le_mul — Mathlib

English

If f and g tend to c and d respectively, then Prod.map f g tends to c.coprod d from a.coprod b.

Русский

Если f и g стремятся к c и d соответственно, то Prod.map f g стремится к c.coprod d из a.coprod b.

LaTeX

$$$ \text{prodMap_coprod} \;: \text{Tendsto } (Prod.map f g) (a.coprod b) (c.coprod d) $$$

Lean4

theorem mul_le_mul [MulZeroClass β] [Preorder β] [PosMulMono β] [MulPosMono β] {l : Filter α} {f₁ f₂ g₁ g₂ : α → β}
    (hf : f₁ ≤ᶠ[l] f₂) (hg : g₁ ≤ᶠ[l] g₂) (hg₀ : 0 ≤ᶠ[l] g₁) (hf₀ : 0 ≤ᶠ[l] f₂) : f₁ * g₁ ≤ᶠ[l] f₂ * g₂ := by
  filter_upwards [hf, hg, hg₀, hf₀] with x using _root_.mul_le_mul

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