ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A nonidentity projection of the plane has determinant zero and is not invertible

Example

Over $R$ , the projection $T (x, y) = (x, 0)$ has determinant $0$ and is not invertible.

Facts & Assumptions

Given: $T : R^{2} \to R^{2}$ , $T (x, y) = (x, 0)$ .

[F1]

$R$ is a field (The reals form a field).

[F2]

An operator determinant is the determinant of any representing matrix (The determinant of an endomorphism of a finite-dimensional vector space: its matrix determinant in an ordered basis in positive dimension, and $1$ on the zero space).

[L1]

A finite-dimensional operator over a field is invertible exactly when its determinant is nonzero (A finite-dimensional linear operator over a field is invertible if and only if its determinant is nonzero).

Verification

technique · direct

1.1

In the standard ordered basis, $[T] = (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array})$ , so [F2] gives $\det (T) = 1 \cdot 0 - 0 \cdot 0 = 0$ .

F1F2algebra

1.2

Directly, $(0, 1)$ is a nonzero vector in the kernel and the image is ${(x, 0) : x \in R}$ , so $T$ is neither injective nor surjective.

givenalgebra

2.1

The invertibility criterion [L1] agrees with step 1.2 because the determinant computed in step 1.1 is zero.

step 1.1step 1.2L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 68 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.