TheoremStatement: AI-adaptedProof: 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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$C$ is the real coordinate plane, with coordinate arithmetic

Statement

The map $Φ : C ⟶ R^{2}, Φ (a + b i) = (a, b),$ is a bijection. Under it, $Φ ((a + b i) + (u + v i)) = (a + u, b + v)$ and $Φ ((a + b i) (u + v i)) = (a u - b v, a v + b u) .$

Facts & Assumptions

Given: The complex field in its quotient construction.

[F1]

Every complex number has a unique form $a + b i$ , and addition and multiplication have the displayed coordinate formulas ( $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ ).

Proof

technique · direct

1.1

Uniqueness in [F1] makes $Φ$ well-defined and injective.

1.2

For every $(a, b) \in R^{2}$ , the complex number $a + b i$ maps to $(a, b)$ , so $Φ$ is surjective.

2.1

Applying $Φ$ to the addition formula in [F1] gives the first coordinate identity.

F1step 1.1

3.1

Applying $Φ$ to the multiplication formula in [F1] gives the second coordinate identity.

F1step 1.1∎

Depends on

$\mathbb C=\mathbb R[x]/(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)/(a^2+b^2)$

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: 21 results over 5 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

R. K. Srivastava, Complex Analysis lecture notes (standard reference, not scraped)

CC is the real coordinate plane, with coordinate arithmetic