THM0019canonicalv1No Ring Negation
No function neg: tau-Idx -> tau-Idx satisfies n + neg(n) = 0 for all n. tau-Idx is a commutative semiring but not a ring.
Payload
No Ring Negation
No function neg: tau-Idx -> tau-Idx satisfies n + neg(n) = 0 for all n. tau-Idx is a commutative semiring but not a ring.
No Ring Negation
Summary
No function neg: tau-Idx -> tau-Idx satisfies n + neg(n) = 0 for all n. tau-Idx is a commutative semiring but not a ring.
Statement
%
\label{thm:no-ring-negation}
There exists no function
$\mathrm{neg} : \tau\text{-Idx} \to \tau\text{-Idx}$
such that $\underline{n} + \mathrm{neg}(\underline{n}) = \underline{0}$
for all $\underline{n} \in \tau$-Idx.
Proof / Justification
Suppose such a function exists.
Apply it to $\underline{1}$:
$\underline{1} + \mathrm{neg}(\underline{1}) = \underline{0}$.
But Theorem~\ref{thm:no-additive-inverse}
asserts that no such $\mathrm{neg}(\underline{1})$ exists,
since $\underline{1} > \underline{0}$.
Contradiction.
Source Context
- Registry source:
book-01.jsonlline 98 - Manuscript source:
2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch11-swap-add-mul.texlines 299-305
Lean / Formalization Notes
- Formalization:
formalized - Module:
TauLib.BookI.Denotation.Structural - Name:
Tau.Denotation.tauIdx_no_ring_negation
Dependencies
- Canonical: I.T14
Related Results
Generated by later projection phases.
Related Publications
Generated by later projection phases.
Revision Notes
- 2026-04-24: Initial pilot migration.
Identifiers
Aliases & legacy IDs
I.T15no-ring-negationthm:no-ring-negationRelease lines
corpus_v3_workingcorpus_v2Relations
Appears in (1)
Sources
Version & History
Status disclaimer
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