feat(RingTheory/HopfAlgebra): prove antipode is antihomomorphism

[kim-em]

This PR proves that the antipode of a Hopf algebra is an antihomomorphism:
antipode (a * b) = antipode b * antipode a.

The proof uses the convolution algebra structure on (A ⊗ A) →ₗ[R] A and shows that S ∘ μ and μ ∘ (S ⊗ S) ∘ comm are both convolution inverses of μ, hence they must be equal by uniqueness of inverses.

This resolves a TODO listed in the file header.

🤖 Prepared with Claude Code

[github-actions]

PR summary cb53e58ea4

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.RingTheory.HopfAlgebra.Basic	1080	1085	+5 (+0.46%)

Import changes for all files

Files	Import difference
3 files Mathlib.Algebra.Category.HopfAlgCat.Monoidal Mathlib.RingTheory.HopfAlgebra.GroupLike Mathlib.RingTheory.HopfAlgebra.TensorProduct	2
Mathlib.Algebra.Category.HopfAlgCat.Basic	3
Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra	4
Mathlib.RingTheory.HopfAlgebra.Basic	5

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Declarations diff

+ antipode_mul

You can run this locally as follows

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

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/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).