Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Geometry and linear algebra

Pythagorean Theorem

For two vectors in a real inner-product space, the square of the length of their sum equals the sum of their squared lengths if and only if their real inner product is zero. For nonzero vectors, this is the familiar right-angle condition.

Exact theorem

Exact Mathlib statement

theorem norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero {F : Type*} [SeminormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F) : ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ inner ℝ x y = 0

The theorem at a glance

Pythagorean theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

For real inner-product-space vectors, the Pythagorean norm identity is equivalent to orthogonality. Explanatory diagram.
Detailed visual description

The poster foregrounds the two-way equivalence between a squared-norm sum and a zero real inner product, while separating that vector statement from a direct claim about triangle side lengths.

Statement structure

Statement and scope

Statement map for Pythagorean TheoremIn a real inner-product space, the squared-norm identity for a vector sum holds exactly when the summands are orthogonal. Claim boundary: This target indexes Mathlib's real inner-product-space equivalence: the squared norm of x+y is the sum of squared norms exactly when x and y are orthogonal. It is a vector identity, not a direct theorem about side lengths of a Euclidean triangle. The pinned upstream declaration is norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingIn a real inner-productspace, the squared-normidentity for a vectorsum holds exactly whenthe summands areorthogonal.Claim boundaryThis target indexesMathlib's realinner-product-spaceequivalence: the squarednorm of x+y is the sumof squared norms exactlywhen x and y areorthogonal. It is avector identity, not adirect theorem aboutside lengths of aEuclidean triangle.Pinned declarationmathlib ·norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zeroStatement map for Pythagorean TheoremIn a real inner-product space, the squared-norm identity for a vector sum holds exactly when the summands are orthogonal. Claim boundary: This target indexes Mathlib's real inner-product-space equivalence: the squared norm of x+y is the sum of squared norms exactly when x and y are orthogonal. It is a vector identity, not a direct theorem about side lengths of a Euclidean triangle. The pinned upstream declaration is norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingIn a real inner-productspace, the squared-normidentity for a vectorsum holds exactly whenthe summands areorthogonal.Claim boundaryThis target indexesMathlib's realinner-product-spaceequivalence: the squarednorm of x+y is the sumof squared norms exactlywhen x and y areorthogonal. It is avector identity, not adirect theorem aboutside lengths of aEuclidean triangle.Pinned declarationmathlib ·norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Orthogonality is exactly the condition behind the squared-norm sum identity. Explanatory scientific diagram.
Detailed visual description

A right-angle vector construction and three area squares connect the norm of a vector sum with the norms of its two perpendicular summands.

Why it matters

A mathematical landmark

The Pythagorean relation is a defining bridge between distance and orthogonality. Mathlib's selected equivalence exposes the inner-product structure behind the familiar right-triangle identity.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed pageCurrent public presentation reviewed
Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's real inner-product-space equivalence: the squared norm of x+y is the sum of squared norms exactly when x and y are orthogonal. It is a vector identity, not a direct theorem about side lengths of a Euclidean triangle.

Source and local evidence

Where the theorem comes from

Existing declaration
norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.pythagorean-theorem.v001
Source
Open the pinned upstream reference