card_le_card_mul_left₀ — Mathlib

English

In a cancellative left-mivial semigroup with zero, if a ∈ s and a ≠ 0, then the cardinality of t does not exceed the cardinality of s·t: |t| ≤ |s·t|.

Русский

В конечном наборе Finset в условиях левостороннего отменяемого нуля, если a∈s и a≠0, то |t| ≤ |s·t|.

LaTeX

$$$|t| \le |s \cdot t| \quad\text{given } a\in s,\ a\neq 0 \text{ and } \alpha\text{ left-cancelling with zero.}$$$

Lean4

theorem card_le_card_mul_left₀ [IsLeftCancelMulZero α] (has : a ∈ s) (ha : a ≠ 0) : #t ≤ #(s * t) :=
  card_le_card_mul_left_of_injective has (mul_right_injective₀ ha)

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