feat(Order/UpperLower/Closure): the upper and lower closures of a maximal antichain cover the order

[AntoineduFresne]

I add IsMaxAntichain.upperClosure_union_lowerClosure. It says that the upper and lower closures of a maximal antichain cover the whole order.

I put it right after ordConnected_iff_upperClosure_inter_lowerClosure, next to upperClosure_inter_lowerClosure, which seems the right spot.

The lemma is about antichains, so Mathlib/Order/Antichain.lean would have looked like the natural file to modify, but upperClosure is defined in Mathlib/Order/UpperLower/Closure.lean, which imports Mathlib.Order.Interval.Set.OrdConnected, which imports Mathlib.Order.Antichain. So to avoid an import loop I put it in Closure.lean.

I split this out of #40741, where a reviewer suggested it. Nothing in that PR uses the lemma.

[github-actions]

Welcome new contributor!

Thank you for contributing to Mathlib! If you haven't done so already, please review our contribution guidelines, as well as the style guide and naming conventions. In particular, we kindly remind contributors that we have guidelines regarding the use of AI when making pull requests.

We use a review queue to manage reviews. If your PR does not appear there, it is probably because it is not successfully building (i.e., it doesn't have a green checkmark), has the awaiting-author tag, or another reason described in the Lifecycle of a PR. The review dashboard has a dedicated webpage which shows whether your PR is on the review queue, and (if not), why.

If you haven't already done so, please come to https://leanprover.zulipchat.com/, introduce yourself, and mention your new PR.

Thank you again for joining our community.

[github-actions]

PR summary aa2bd7c434

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ IsMaxAntichain.upperClosure_union_lowerClosure

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit aa2bd7c).

• +1 new declarations
• −0 removed declarations

+IsMaxAntichain.upperClosure_union_lowerClosure

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit aa2bd7c434
Reference commit 639923353f

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[tdwag123]

easy