Theorems · Theorem · linear algebra
Submodule.span_eq_bot
∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {s : Set M},
Submodule.span R s = ⊥ ↔ ∀ x ∈ s, x = 0- Defined in
- Mathlib.LinearAlgebra.Span.Defs
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 25 from the axioms · uses propext, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
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.
- Setstatement and proof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Submodulestatement · cited by 7,192
- Bot.botstatement and proof · cited by 4,720
- Submodule.spanstatement and proof · cited by 1,504
- Submodule.subset_spanproof · cited by 234
- Submodule.span_leproof · cited by 164
- eq_bot_iffproof · cited by 159
- Submodule.mem_botproof · cited by 55
Cited by3
Results whose statement or proof uses this declaration.
- FractionalIdeal.spanSingleton_eq_zero_iffproof · cited by 3
- LieModule.ideal_oper_maxTrivSubmodule_eq_botproof · cited by 2
- KaehlerDifferential.ker_map_of_surjectiveproof · cited by 1