mk_mul_of_mem — Mathlib

English

The preimage of the image of a subset s under the quotient map equals the union, over elements x of N, of the left- translates of s by x.

Русский

Образ предобраза под действием потока деления: прееобраз пред image mk равен объединению по x ∈ N множителей s слева.

LaTeX

$$$\operatorname{preimage}(\text{QuotientGroup.mk}, \text{image}(\text{QuotientGroup.mk} \, s)) = \bigcup_{x \in N} (\cdot x)^{-1}(s)$$

Lean4

@[to_additive (attr := simp)]
theorem mk_mul_of_mem (a : α) (hb : b ∈ s) : (mk (a * b) : α ⧸ s) = mk a := by
  rwa [QuotientGroup.eq, mul_inv_rev, inv_mul_cancel_right, s.inv_mem_iff]

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