Theorems · Theorem · commutative algebra
two_smul
∀ (R : Type u_1) {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (x : M),
2 • x = x + x- Defined in
- Mathlib.Algebra.Module.Defs
- Cited by
- 57 results in Mathlib
- Foundations
- Depth 19 from the axioms · uses propext
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- one_smulproof · cited by 1,374
- add_smulproof · cited by 204
- one_add_one_eq_twoproof · cited by 65
Cited by57
Results whose statement or proof uses this declaration.
- realPart_add_I_smul_imaginaryPartproof · cited by 10
- RootPairing.root_sub_root_mem_of_pairingIn_posproof · cited by 7
- QuadraticMap.associated_eq_self_applyproof · cited by 7
- midpoint_eq_smul_addproof · cited by 6
- EuclideanGeometry.reflection_apply'proof · cited by 6
- hasStrictFDerivAt_norm_sqproof · cited by 6
- LieAlgebra.IsKilling.exists_isSl2Triple_of_weight_isNonZeroproof · cited by 5
- RootPairing.pairing_reflectionPerm_self_rightproof · cited by 4
- QuadraticMap.polar_selfproof · cited by 3
- QuadraticMap.radical_eq_ker_polarBilinproof · cited by 3
- RootPairing.pairing_reflectionPerm_self_leftproof · cited by 3
- Submodule.reflection_eq_self_iffproof · cited by 3