mem
plain-language theorem explainer
The lemma equates the predicate that x is algebraic over the subfield generated by φ with direct membership in that subfield. Modelers verifying closure properties inside Recognition Science algebraic structures would cite the equivalence when simplifying membership tests. The proof is immediate reflexivity on the definition of PhiClosed.
Claim. Let $φ, x ∈ ℝ$. Then $x$ is algebraic over the subfield generated by $φ$ using field operations if and only if $x$ belongs to the subfield generated by $φ$.
background
In RecogSpec.Core the subfield phiSubfield φ is the smallest subfield of ℝ containing φ, obtained as Subfield.closure({φ}). PhiClosed φ x is defined to hold precisely when x lies in this subfield, i.e., when x can be reached from φ by the field operations of addition, multiplication and inversion. The present lemma therefore records the definitional unfolding of that predicate.
proof idea
The proof is a one-line wrapper that applies reflexivity to the defining equality PhiClosed φ x ↔ x ∈ phiSubfield φ.
why it matters
The equivalence feeds the membership check inside isMinimalIn of the Atomicity module. It supplies the basic interface step needed whenever Recognition Science arguments must confirm that a real number remains inside the phi-generated subfield, a structure that appears throughout the phi-ladder and forcing-chain constructions.
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