Theorems · Theorem · group theory
Commute.isUnit_mul_iff
∀ {M : Type u_1} [inst : Monoid M] {a b : M}, Commute a b → (IsUnit (a * b) ↔ IsUnit a ∧ IsUnit b)- Defined in
- Mathlib.Algebra.Group.Commute.Units
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 15 from the axioms · uses propext
- Assumes
- Monoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Monoidstatement and proof · cited by 3,887
- Unitsproof · cited by 2,804
- Units.valproof · cited by 1,966
- IsUnitstatement and proof · cited by 1,602
- Commutestatement and proof · cited by 639
- Units.isUnitproof · cited by 116
- IsUnit.mulproof · cited by 32
- Units.leftOfMulproof · cited by 1
- Units.rightOfMulproof · cited by 1
Cited by9
Results whose statement or proof uses this declaration.
- spectrum.subset_polynomial_aevalproof · cited by 4
- spectrum.map_polynomial_aeval_of_degree_posproof · cited by 2
- IsUnit.natCast_factorial_of_leproof · cited by 2
- Ring.mul_inverse_rev'proof · cited by 2
- spectrum.exp_mem_expproof · cited by 1
- spectrum.inv_mem_resolventSetproof · cited by 1
- IsLocalizedModule.of_exists_mul_memproof · cited by 1
- IsUnit.natCast_factorial_iff_of_charPproof · cited by 0
- isUnit_mul_self_iffproof · cited by 0