Corpus v3 · Definition
cid001047
DEF0045canonicalv1tau-Membership Relation
A in_tau B iff A | B (in the multiplicative structure on tau-Idx). Membership earned from divisibility; decidable, reflexive, antisymmetric. No ZFC axioms imported.
Payload
tau-Membership Relation
| A in_tau B iff A | B (in the multiplicative structure on tau-Idx). Membership earned from divisibility; decidable, reflexive, antisymmetric. No ZFC axioms imported. |
tau-Membership Relation
Summary
| A in_tau B iff A | B (in the multiplicative structure on tau-Idx). Membership earned from divisibility; decidable, reflexive, antisymmetric. No ZFC axioms imported. |
Statement
%
\label{def:tau-membership}
For $\underline{a}, \underline{b} \in \tau\text{-Idx}$,
we define the \textbf{$\tau$-membership relation} by:
\[
\boxed{%
\underline{a} \in_\tau \underline{b}
\quad\iff\quad
\underline{a} \mid \underline{b}.}
\]
That is, $\underline{a}$ is ``an element of'' $\underline{b}$
if and only if $\underline{a}$ divides $\underline{b}$
in the multiplicative structure on $\tau$-Idx.
Proof / Justification
This item is definitional. No manuscript proof is required.
Source Context
- Registry source:
book-01.jsonlline 77 - Manuscript source:
2nd-edition/book-i-categorical-foundations/02_mainmatter/part08/ch32-membership-divisibility.texlines 37-51
Lean / Formalization Notes
- Formalization:
formalized - Module:
TauLib.BookI.Sets.Membership - Name:
Tau.Sets.tau_mem
Dependencies
- Canonical: I.D19a, I.T09, I.T04
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.D31tau-membership-relationdef:tau-membershipRelease lines
corpus_v3_workingcorpus_v2Relations
Appears in (1)
Sources
Version & History
Status disclaimer
A Corpus Item page reports the program's current internal record for this item. It does not imply external verification, scientific consensus, or final proof unless explicitly stated. Read it together with its dependencies, formalization status, and the program's overall stance.