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feat(Algebra/GroupWithZero/Divisibility): add mul_dvd_left_iff_isUnit #39763

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feat(Algebra/GroupWithZero/Divisibility): add mul_dvd_left_iff_isUnit #39763

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add `mul_dvd_left_iff_isUnit`
NoahW314 May 24, 2026
1dbc557
Update Divisibility.lean
NoahW314 May 24, 2026
3ce69d5
Merge branch 'master' into mul_dvd_isUnit
NoahW314 Jun 20, 2026
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9 changes: 9 additions & 0 deletions Mathlib/Algebra/GroupWithZero/Divisibility.lean
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Expand Up @@ -172,6 +172,15 @@ lemma pow_dvd_pow_iff (ha₀ : a ≠ 0) (ha : ¬IsUnit a) : a ^ n ∣ a ^ m ↔
apply pow_ne_zero m ha₀
· apply pow_dvd_pow

lemma mul_dvd_left_iff_isUnit (ha0 : a ≠ 0) : a * b ∣ a ↔ IsUnit b := by
nth_rw 2 [← mul_one a]
rw [mul_dvd_mul_iff_left ha0]
exact isUnit_iff_dvd_one.symm

lemma mul_dvd_right_iff_isUnit (ha0 : a ≠ 0) : b * a ∣ a ↔ IsUnit b := by
rw [mul_comm]
exact mul_dvd_left_iff_isUnit ha0

end CancelCommMonoidWithZero

section GroupWithZero
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