Theorems · Theorem · commutative algebra

Subsemiring.closure_induction

∀ {R : Type u} [inst : NonAssocSemiring R] {s : Set R} {p : (x : R) → x ∈ Subsemiring.closure s → Prop},
  (∀ (x : R) (hx : x ∈ s), p x ⋯) →
    p 0 ⋯ →
      p 1 ⋯ →
        (∀ (x y : R) (hx : x ∈ Subsemiring.closure s) (hy : y ∈ Subsemiring.closure s), p x hx → p y hy → p (x + y) ⋯) →
          (∀ (x y : R) (hx : x ∈ Subsemiring.closure s) (hy : y ∈ Subsemiring.closure s),
              p x hx → p y hy → p (x * y) ⋯) →
            ∀ {x : R} (hx : x ∈ Subsemiring.closure s), p x hx

An induction principle for closure membership. If p holds for 0, 1, and all elements of s, and is preserved under addition and multiplication, then p holds for all elements of the closure of s.

Defined in

Mathlib.Algebra.Ring.Subsemiring.Basic

Cited by

3 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NonAssocSemiring