Theorems · Theorem · linear algebra

LinearPMap.mem_domain_sSup_iff

∀ {R : Type u_1} {S : Type u_2} [inst : Ring R] [inst_1 : Ring S] {σ : R →+* S} {E : Type u_4} [inst_2 : AddCommGroup E]
  [inst_3 : Module R E] {F : Type u_5} [inst_4 : AddCommGroup F] [inst_5 : Module S F] {c : Set (E →ₛₗ.[σ] F)},
  c.Nonempty →
    ∀ (hc : DirectedOn (fun x1 x2 => x1 ≤ x2) c) {x : E}, x ∈ (LinearPMap.sSup c hc).domain ↔ ∃ f ∈ c, x ∈ f.domain

Defined in

Mathlib.LinearAlgebra.LinearPMap

Cited by

2 results in Mathlib

Foundations

Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingRingAddCommGroupModuleAddCommGroupModule