commute_of_mem_adjoin_of_forall_mem_commute

English

If c ∈ adjoin R {b} and a commutes with b, then a commutes with c.

Русский

Если c ∈ adjoin_R{b} и a commute c, то a commute с c.

LaTeX

$$$\\text{If } c \\in \\operatorname{adjoin}_R\\{b\\} \\text{ and } \\operatorname{Commute}(a,b), \\ \\operatorname{Commute}(a,c)$$$

Lean4

theorem commute_of_mem_adjoin_of_forall_mem_commute {a b : A} {s : Set A} (hb : b ∈ adjoin R s)
    (h : ∀ b ∈ s, Commute a b) : Commute a b := by
  induction hb using adjoin_induction with
  | mem x hx => exact h x hx
  | algebraMap r => exact commutes r a |>.symm
  | add y z _ _ hy hz => exact hy.add_right hz
  | mul y z _ _ hy hz => exact hy.mul_right hz

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