Theorems · Theorem · linear algebra

Module.nontrivial_of_finrank_eq_succ

∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [Nontrivial R] {n : ℕ},
  Module.finrank R M = n.succ → Nontrivial M

A finite-dimensional space is nontrivial if it has finrank equal to the successor of a natural number.

Defined in: Mathlib.LinearAlgebra.Dimension.Finite
Cited by: 4 results in Mathlib
Foundations: Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Semiring AddCommMonoid Module Nontrivial

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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
Nontrivialstatement and proof · cited by 2,416
Module.finrankstatement and proof · cited by 1,770
Module.nontrivial_of_finrank_posproof · cited by 7

Cited by4

Results whose statement or proof uses this declaration.

is_simple_module_of_finrank_eq_oneproof · cited by 2
Orientation.det_rotationproof · cited by 1
WittVector.isocrystal_classificationproof · cited by 0
Orientation.exists_linearIsometryEquiv_eq_of_det_posproof · cited by 0