mul — Mathlib

English

Equivalent reformulations of the mem_nullSubmodule_iff statement.

Русский

Эквивалентные формулировки mem_nullSubmodule_iff.

LaTeX

$$$x \in nullSubmodule 𝕜 E \iff \|x\| = 0$$$

Lean4

@[to_additive]
theorem mul (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s * t) :=
  by
  obtain ⟨Rs, hRs⟩ : ∃ R, ∀ x ∈ s, ‖x‖ ≤ R := hs.exists_norm_le'
  obtain ⟨Rt, hRt⟩ : ∃ R, ∀ x ∈ t, ‖x‖ ≤ R := ht.exists_norm_le'
  refine isBounded_iff_forall_norm_le'.2 ⟨Rs + Rt, ?_⟩
  rintro z ⟨x, hx, y, hy, rfl⟩
  exact norm_mul_le_of_le' (hRs x hx) (hRt y hy)

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