slope_comm
∀ {k : Type u_1} {E : Type u_2} {PE : Type u_3} [inst : Field k] [inst_1 : AddCommGroup E] [inst_2 : Module k E]
[inst_3 : AddTorsor E PE] (f : k → PE) (a b : k), slope f a b = slope f b a- Defined in
- Mathlib.LinearAlgebra.AffineSpace.Slope
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- AddCommGroupstatement and proof · cited by 12,871
- Fieldstatement and proof · cited by 7,404
- AddTorsorstatement and proof · cited by 1,657
- VSub.vsubproof · cited by 817
- neg_smulproof · cited by 306
- neg_subproof · cited by 272
- smul_negproof · cited by 181
- slopestatement and proof · cited by 147
- neg_vsub_eq_vsub_revproof · cited by 87
- neg_invproof · cited by 11
Cited by9
Results whose statement or proof uses this declaration.
- ConvexOn.slope_le_of_hasDerivWithinAt_Iioproof · cited by 5
- StrictConvexOn.slope_lt_of_hasDerivWithinAt_Iioproof · cited by 4
- MonotoneOn.slope_nonnegproof · cited by 3
- ConvexOn.isMinOn_of_leftDeriv_nonpos_of_rightDeriv_nonnegproof · cited by 2
- lineMap_slope_lineMap_slope_lineMapproof · cited by 1
- slope_pos_iff_gtproof · cited by 1
- StrictMonoOn.slope_posproof · cited by 1
- ConvexOn.monotoneOn_leftDerivproof · cited by 0
- ConvexOn.monotoneOn_rightDerivproof · cited by 0