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[Merged by Bors] - feat: Add mul_inv and inv_mul versions of div_le_one_of_le #10597

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[Merged by Bors] - feat: Add mul_inv and inv_mul versions of div_le_one_of_le #10597

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feat: `mul_inv_le_one_of_le` is an `a * b⁻¹` `div_le_one_of_le`
girving Feb 15, 2024
e567f88
Add the symmetric inv_mul version
girving Feb 15, 2024
563bd18
Remove bad docstrings
girving Feb 20, 2024
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8 changes: 8 additions & 0 deletions Mathlib/Algebra/Order/Field/Basic.lean
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Expand Up @@ -243,6 +243,14 @@ theorem div_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : a / b ≤ 1 :=
div_le_of_nonneg_of_le_mul hb zero_le_one <| by rwa [one_mul]
#align div_le_one_of_le div_le_one_of_le

/-- `a * b⁻¹` version of `div_le_one_of_le` -/
lemma mul_inv_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : a * b⁻¹ ≤ 1 := by
simpa only [← div_eq_mul_inv] using div_le_one_of_le h hb

/-- `b⁻¹ * a` version of `div_le_one_of_le` -/
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lemma inv_mul_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : b⁻¹ * a ≤ 1 := by
simpa only [← div_eq_inv_mul] using div_le_one_of_le h hb

/-!
### Bi-implications of inequalities using inversions
-/
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