feat(Analysis/Normed): mapping a ball with smul

[wwylele]

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[github-actions]

PR summary 090daf0bfb

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff

+ Metric.smul_image_ball
+ Metric.smul_image_closedBall
+ Metric.smul_image_sphere

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.

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No changes to strong technical debt.

No changes to weak technical debt.

[Copilot]

Pull request overview

This PR adds lemmas in Mathlib/Analysis/Normed/MulAction.lean describing how scalar multiplication by a nonzero real number maps metric balls, closed balls, and spheres in a seminormed real module equipped with NormSMulClass.

Changes:

• Add Metric.smul_image_ball identifying (s • ·) '' Metric.ball x ε with a ball of radius |s| * ε.
• Add Metric.smul_image_closedBall identifying (s • ·) '' Metric.closedBall x ε with a closed ball of radius |s| * ε.
• Add Metric.smul_image_sphere identifying (s • ·) '' Metric.sphere x ε with a sphere of radius |s| * ε.

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