feat(Geometry/Manifold): lemmas relating writtenInExtChartAt and extChartAt

[pepamontero]

This PR adds three lemmas relating writtenInExtChartAt and extChartAt to ExtChartAt.lean:

• writtenInExtChartAt_apply_extChartAt,
• extChartAt_symm_writtenInExtChartAt and
• extChartAt_symm_writtenInExtChartAt_extChartAt.

The first two are used to golf some proofs; some of them did the same reasoning in writtenInExtChartAt_apply_extChartAt using a have. The others previously unfolded writtenInExtChartAt by hand and relied on simp only [*].

The third one is not used here, but it could make a proof nicer in #34005.

AI disclosure: AI was used to scan places where these lemmas could be reused, as they were originally only written to use in #34005.

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[github-actions]

PR summary 51a326a7e7

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 (regex)

+ extChartAt_symm_writtenInExtChartAt
+ extChartAt_symm_writtenInExtChartAt_extChartAt
+ writtenInExtChartAt_apply_extChartAt

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 51a326a).

• +3 new declarations
• −0 removed declarations

+extChartAt_symm_writtenInExtChartAt
+extChartAt_symm_writtenInExtChartAt_extChartAt
+writtenInExtChartAt_apply_extChartAt

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No changes to strong technical debt.
No changes to weak technical debt.

Current commit 51a326a7e7
Reference commit 6ebcdee77a

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.py 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).