Public Mathlib landmark · existing upstream theorem · read-only
ProofAtlas

Geometry and linear algebra · existing Mathlib theorem

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 the vectors meet at a right angle.

Four separate status axes

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed page2 of 3 presentation reviews recorded
Accepted Atlas resultNot recorded for the preferred artifact

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

Formal orientation

Statement map

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. The pinned upstream declaration is norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero. The exact checked statement is 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.Mathematical readingIn a real inner-productspace, the squared-normidentity for a vectorsum holds exactly whenthe summands areorthogonal.Pinned declarationmathlib ·norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zeroExact checked formtheoremnorm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero{F : Type*}[SeminormedAddCommGroupF] [InnerProductSpace ℝF] (x y : F) : ‖x + y‖ *‖x + y‖ = ‖x‖ * ‖x‖ +‖y‖ * ‖y‖ ↔ inner ℝ x y= 0Statement map for Pythagorean TheoremIn a real inner-product space, the squared-norm identity for a vector sum holds exactly when the summands are orthogonal. The pinned upstream declaration is norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero. The exact checked statement is 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.Mathematical readingIn a real inner-productspace, the squared-normidentity for a vectorsum holds exactly whenthe summands areorthogonal.Pinned declarationmathlib ·norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zeroExact checked formtheoremnorm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero{F : Type*}[SeminormedAddCommGroupF] [InnerProductSpace ℝF] (x y : F) : ‖x + y‖ *‖x + y‖ = ‖x‖ * ‖x‖ +‖y‖ * ‖y‖ ↔ inner ℝ x y= 0

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

The theorem at a glance

Pythagorean theorem at a glance

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

For real inner-product-space vectors, the Pythagorean norm identity is equivalent to orthogonality. Generated explanation only; the exact HTML statement and pinned source are authoritative.
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.

Orthogonality is exactly the condition behind the squared-norm sum identity. Generated explanation only; the exact HTML statement and pinned source are authoritative.
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.

Complementary intuition

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.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

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
Selected evidence declaration text
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

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.

Provenance

Origin and evidence stay separate

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
Preferred checked artifact
artifact.library.mathlib.pythagorean-theorem.v001
Source handling
Verified upstream reference; no mirrored source package