Algebra · easy

Statement, exactly as the model and Lean see it

theorem lg_target {ι : Type u_1} {α : Type u_2} {R : ι → Type u_3} [(i : ι) → SMul α (R i)] {r : α} [∀ (i : ι), Nonempty (R i)] : IsSMulRegular ((i : ι) → R i) r ↔ ∀ (i : ι), IsSMulRegular (R i) r := by

Mathlib declaration: Pi.isSMulRegular_iff
Held-out module: Mathlib.Algebra.Regular.Pi
Banned modules (itself + downstream): 1
Reference proof premises (reachable / held-out): 1 / 0

Traces

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No configuration has attempted this theorem yet.

Reference proof

Mathlib's own source for this declaration, shown for comparison. It may use lemmas the prover is not allowed to use.

@[simp]
theorem isSMulRegular_iff [∀ i, SMul α (R i)] {r : α} [∀ i, Nonempty (R i)] :
    IsSMulRegular (∀ i, R i) r ↔ ∀ i, IsSMulRegular (R i) r :=
  Pi.map_injective