Theorems · Theorem · linear algebra

LinearPMap.image_iff

∀ {R : Type u_1} [inst : Ring R] {E : Type u_4} [inst_1 : AddCommGroup E] [inst_2 : Module R E] {F : Type u_5}
  [inst_3 : AddCommGroup F] [inst_4 : Module R F] {f : E →ₗ.[R] F} {x : E} {y : F} (hx : x ∈ f.domain),
  y = ↑f ⟨x, hx⟩ ↔ (x, y) ∈ f.graph

Defined in: Mathlib.LinearAlgebra.LinearPMap
Cited by: 2 results in Mathlib
Foundations: Depth 31 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Ring AddCommGroup Module AddCommGroup Module

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Cites9

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
RingHom.idstatement and proof · cited by 18,349
AddCommGroupstatement and proof · cited by 12,871
Ringstatement and proof · cited by 7,463
Submodulestatement · cited by 7,192
LinearPMapstatement and proof · cited by 179
LinearPMap.domainstatement and proof · cited by 167
LinearPMap.toFun'statement · cited by 124
LinearPMap.graphstatement · cited by 39

Cited by2

Results whose statement or proof uses this declaration.

LinearPMap.le_of_le_graphproof · cited by 4
LinearPMap.inverse_isClosable_iffproof · cited by 1