CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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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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The number $e$ is the unique $x > 0$ satisfying $\int_{1}^{x} \frac{d t}{t} = 1$

Statement

The number $e$ is the unique $x > 0$ such that

$\int_{1}^{x} \frac{d t}{t} = 1.$

Facts & Assumptions

Given: The published number $e$ and the integral function $L$ .

[F1]

$e : = \exp (1)$ (The real exponential function and the number $e$ by a power series).

[L1]

$E = \exp$ (The integral exponential $E$ is the published exponential function).

[L2]

$L$ is strictly increasing on $(0, \infty)$ (The integral logarithm is continuous and strictly increasing on $(0, \infty)$ ).

[F2]

$E = L^{- 1}$ , so $L (E (y)) = y$ (The integral exponential $E : R \to (0, \infty)$ as the inverse of $L$ ).

Proof

technique · direct

1.1

By [F1] and [L1], $e = E (1)$ . The inverse identity [F2] gives $L (e) = L (E (1)) = 1$ .

F1L1F2algebra

2.1

Strict increase [L2] makes $L$ injective. Hence if $x > 0$ also satisfies $L (x) = 1 = L (e)$ , then $x = e$ .

L2step 1.1

3.1

The defining integral for $L$ converts steps 1.1 and 2.1 into the stated existence and uniqueness claim.

step 1.1step 2.1∎

Depends on

Used by

Integral bounds alone give 2<e<4; the sharper published bound is 2<e<3 Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 72 results over 18 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.

Sources

OpenStax, Calculus Volume 1, Section 6.7 (standard reference, not scraped)

The number ee is the unique x>0x>0 satisfying ∫1xdtt=1∫1x​tdt​=1