toComplex_im
plain-language theorem explainer
The theorem records that the imaginary part of the transported complex equals the transport of its imaginary component. Researchers verifying componentwise properties of the recovered-to-standard complex equivalence would cite it during simplification passes. The proof is immediate reflexivity from the componentwise definition of the transport map.
Claim. Let $z$ be a complex number constructed as a pair of recovered reals. Then the imaginary part of its image under the transport map to the standard complex numbers equals the standard real obtained by transporting the imaginary component of $z$.
background
LogicComplex is the structure whose fields are a recovered real part and a recovered imaginary part. The transport map toComplex sends such a pair to a standard complex number by applying the toReal conversion to each field separately. The toReal conversion itself maps a recovered real to a standard real via the equivalence CompareReals.compareEquiv on its underlying value.
proof idea
The proof is a one-line reflexivity wrapper. It holds by the definition of toComplex, which builds the standard complex number directly from the pair of toReal applications to the input components.
why it matters
The result is part of the carrier-level equivalence between recovered complexes and Mathlib complexes, allowing later modules to invoke standard complex analysis without redeveloping it. It supplies a simp lemma for imaginary-part extraction after transport. No downstream uses are recorded yet; the declaration closes a basic projection identity needed for the overall real-to-complex recovery in the foundation layer.
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