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feat(GroupTheory/Coset/Basic): products, leftRel and rightRel #17585

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feat(GroupTheory/Coset/Basic): products, leftRel and rightRel #17585

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28 changes: 28 additions & 0 deletions Mathlib/GroupTheory/Coset/Basic.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -262,6 +262,20 @@ instance leftRelDecidable [DecidablePred (· ∈ s)] : DecidableRel (leftRel s).
rw [leftRel_eq]
exact ‹DecidablePred (· ∈ s)› _

@[to_additive]
lemma leftRel_prod {β : Type*} [Group β] (s' : Subgroup β) :
leftRel (s.prod s') = (leftRel s).prod (leftRel s') := by
refine Setoid.ext fun x y ↦ ?_
rw [Setoid.prod_apply]
simp_rw [leftRel_apply]
rfl

@[to_additive]
lemma leftRel_pi {ι : Type*} {β : ι → Type*} [∀ i, Group (β i)] (s' : ∀ i, Subgroup (β i)) :
leftRel (Subgroup.pi Set.univ s') = @piSetoid _ _ fun i ↦ leftRel (s' i) := by
refine Setoid.ext fun x y ↦ ?_
simp [Setoid.piSetoid_apply, leftRel_apply, Subgroup.mem_pi]

/-- `α ⧸ s` is the quotient type representing the left cosets of `s`.
If `s` is a normal subgroup, `α ⧸ s` is a group -/
@[to_additive "`α ⧸ s` is the quotient type representing the left cosets of `s`. If `s` is a normal
Expand Down Expand Up @@ -304,6 +318,20 @@ instance rightRelDecidable [DecidablePred (· ∈ s)] : DecidableRel (rightRel s
rw [rightRel_eq]
exact ‹DecidablePred (· ∈ s)› _

@[to_additive]
lemma rightRel_prod {β : Type*} [Group β] (s' : Subgroup β) :
rightRel (s.prod s') = (rightRel s).prod (rightRel s') := by
refine Setoid.ext fun x y ↦ ?_
rw [Setoid.prod_apply]
simp_rw [rightRel_apply]
rfl

@[to_additive]
lemma rightRel_pi {ι : Type*} {β : ι → Type*} [∀ i, Group (β i)] (s' : ∀ i, Subgroup (β i)) :
rightRel (Subgroup.pi Set.univ s') = @piSetoid _ _ fun i ↦ rightRel (s' i) := by
refine Setoid.ext fun x y ↦ ?_
simp [Setoid.piSetoid_apply, rightRel_apply, Subgroup.mem_pi]

/-- Right cosets are in bijection with left cosets. -/
@[to_additive "Right cosets are in bijection with left cosets."]
def quotientRightRelEquivQuotientLeftRel : Quotient (QuotientGroup.rightRel s) ≃ α ⧸ s where
Expand Down
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