feat(Topology/Group/Profinite): Profinite group is limit of finite group

[Thmoas-Guan]

Prove that any profinite group is limit of finite groups.

---

• depends on: [Merged by Bors] - feat(Topology/Group/Profinite): limits in the category of profinite groups #16976
• depends on: [Merged by Bors] - feat(Topology/Group): Open normal subgroup #16980
• depends on: feat(Topology/Group): Continuous isomorphism #16991

[github-actions]

PR summary 32076229bf

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits	1428

---

Declarations diff

+ CanonicalQuotientMap
+ ContinuousAddEquiv
+ ContinuousAddEquivClass
+ ContinuousMulEquiv
+ ContinuousMulEquivClass
+ ContinuousMulEquivClass.toContinuousMulEquiv
+ ContinuousMulEquivClass.toContinuousMulEquiv_injective
+ ContinuousMulEquivLimitQuotientOpenNormalSubgroup
+ QuotientOpenNormalSubgroup
+ apply_eq_iff_eq
+ apply_eq_iff_symm_apply
+ apply_symm_apply
+ bijective
+ canonicalQuotientMap_dense
+ canonicalQuotientMap_injective
+ canonicalQuotientMap_surjective
+ coe_mk
+ coe_refl
+ coe_toHomeomorph_symm
+ coe_toMulEquiv_symm
+ coe_trans
+ comp_symm_eq
+ congr_arg
+ congr_fun
+ continuousMulEquivOfUnique
+ eq_comp_symm
+ eq_symm_apply
+ eq_symm_comp
+ equivLike_inv_eq_symm
+ exist_openNormalSubgroup_sub_open_nhd_of_one
+ ext
+ injective
+ instance (P : ProfiniteGrp) : SmallCategory (OpenNormalSubgroup P)
+ instance : Coe (M ≃ₜ* N) (M ≃ₜ N)
+ instance : CoeFun (M ≃ₜ* N) fun _ ↦ M → N
+ instance : ContinuousMulEquivClass (M ≃ₜ* N) M N
+ instance : EquivLike (M ≃ₜ* N) M N
+ instance : Inhabited (M ≃ₜ* M) := ⟨ContinuousMulEquiv.refl M⟩
+ instance [Mul α] [Mul β] [ContinuousMulEquivClass F α β] : CoeTC F (α ≃ₜ* β)
+ instance {M N} [Unique M] [Unique N] [Mul M] [Mul N]
+ invFun_eq_symm
+ isClosedEmbedding
+ isClosedMap
+ isDenseEmbedding
+ isEmbedding
+ isHomeomorph
+ isInducing
+ isOpenEmbedding
+ isQuotientMap
+ map_eq_one_iff
+ map_mul
+ map_ne_one_iff
+ mk'
+ normal_iInf_normal
+ ofContinuousMulEquivProfiniteGrp
+ refl
+ refl_apply
+ self_comp_symm
+ self_trans_symm
+ surjective
+ symm
+ symm_apply_apply
+ symm_apply_eq
+ symm_comp_eq
+ symm_comp_self
+ symm_symm
+ symm_trans_apply
+ symm_trans_self
+ toEquiv_eq_coe
+ toMulEquiv_eq_coe
+ toMulEquiv_symm
+ totallyDisconnectedSpace
+ trans
+ trans_apply

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.

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• [Merged by Bors] - feat(Topology/Group/Profinite): limits in the category of profinite groups #16976
• [Merged by Bors] - feat(Topology/Group): Open normal subgroup #16980
• feat(Topology/Group): Continuous isomorphism #16991
  By Dependent Issues (🤖). Happy coding!