IsDirectLimit.lean`

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20 declarations, 0 contain sorry, 0 declared as axioms

Is Direct Limit

Material destined for Mathlib.

Imports (1)

Mathlib.Algebra.Colimit.Module

Imported by 3 files

class IsDirectLimit IsDirectLimit f g states that P is the direct limit of the system (M i, f i j hij) line 30 source
Source documentation
IsDirectLimit f g states that P is the direct limit of the system (M i, f i j hij). It requires:
- surj: says every element of P comes from some M i via the canonical map g i.
- inj: says that if two elements of (possibly different) modules M i and M j map to the same element of P, then they become equal after being mapped into a common later module M k.
- compatibility: ensures that the maps g i and f i j hij are compatible with one another.
no axioms what this means
Statement uses: Preorder, LE.le, DirectedSystem
1 instances, coercions and auxiliary constants left out.
```
@[mk_iff] class IsDirectLimit [DirectedSystem M f] : Prop where
…
```
theorem IsDirectLimit.compatibility' line 40 source
no axioms what this means
Statement uses: Preorder, LE.le, DirectedSystem, IsDirectLimit, Eq
Proof uses: IsDirectLimit.compatibility
2 instances, coercions and auxiliary constants left out.
theorem IsDirectLimit.is_injective line 43 source
no axioms what this means
Statement uses: Preorder, LE.le, DirectedSystem, IsDirectLimit, Eq, Exists
Proof uses: IsDirectLimit.inj
2 instances, coercions and auxiliary constants left out.
theorem IsDirectLimit.is_surjective line 48 source
no axioms what this means
Statement uses: Preorder, LE.le, DirectedSystem, IsDirectLimit, Exists, Eq
Proof uses: IsDirectLimit.surj
2 instances, coercions and auxiliary constants left out.
def IsDirectLimit.preimageIndex Documented declaration: preimageIndex line 55 source
Source documentation
Given an element p : P in the direct limit, preimageIndex f g p returns an index in ι such that p is in the image of the map g (preimageIndex f g p) : M (preimageIndex f g p) →ₗ[R] P.

standard axioms: Classical.choice what this means
Statement uses: Preorder, LE.le, DirectedSystem, IsDirectLimit
Proof uses: Exists.choose, Exists, Eq, IsDirectLimit.is_surjective
2 instances, coercions and auxiliary constants left out.
```
noncomputable def preimageIndex (p : P) : ι := (is_surjective f g p).choose
```
Mentions: IsDirectLimit.is_surjective
def IsDirectLimit.preimage Given an element p : P in the direct limit, preimage f g p returns an element of M (preimageIndex f g p) that maps to p under g (preimageIndex f g p) line 61 source
Source documentation
Given an element p : P in the direct limit, preimage f g p returns an element of M (preimageIndex f g p) that maps to p under g (preimageIndex f g p).

standard axioms: Classical.choice what this means
Statement uses: Preorder, LE.le, DirectedSystem, IsDirectLimit, IsDirectLimit.preimageIndex
Proof uses: Exists.choose, Exists, Eq, IsDirectLimit.is_surjective
3 instances, coercions and auxiliary constants left out.
```
noncomputable def preimage (p : P) : M (preimageIndex f g p) :=
…
```
Mentions: IsDirectLimit.preimageIndex
theorem IsDirectLimit.image_preimage line 64 source
standard axioms: Classical.choice what this means
Statement uses: Preorder, LE.le, DirectedSystem, IsDirectLimit, Eq, IsDirectLimit.preimageIndex, IsDirectLimit.preimage
Proof uses: Exists.choose_spec, Exists.choose, Exists, IsDirectLimit.is_surjective
2 instances, coercions and auxiliary constants left out.
def IsDirectLimit.lift Construct a map P₁ → P₂ from a direct limit by sending each element to its preimage in some component M i and applying g₂ i line 74 source
Source documentation
Construct a map P₁ → P₂ from a direct limit by sending each element to its preimage in some component M i and applying g₂ i. Used to define morphisms out of direct limits.

standard axioms: Classical.choice what this means
Statement uses: Preorder, LE.le, DirectedSystem, IsDirectLimit
Proof uses: IsDirectLimit.preimageIndex, IsDirectLimit.preimage
2 instances, coercions and auxiliary constants left out.
```
noncomputable def lift [IsDirectLimit f g₁] (p : P₁) : P₂ :=
…
```
Mentions: IsDirectLimit
theorem IsDirectLimit.lift_of line 78 source
standard axioms: Classical.choice what this means
Statement uses: Preorder, LE.le, DirectedSystem, IsDirectLimit, Eq, IsDirectLimit.lift
Proof uses: id, Exists, IsDirectLimit.preimageIndex, IsDirectLimit.preimage, IsDirectLimit.is_injective, IsDirectLimit.image_preimage, Eq.mpr, congrArg, Eq.symm, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
def IsDirectLimit.Equiv Any two direct limits of the same directed system are isomorphic line 85 source
Source documentation
Any two direct limits of the same directed system are isomorphic.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Preorder, LE.le, DirectedSystem, IsDirectLimit, Equiv
Proof uses: Equiv.mk, IsDirectLimit.lift
4 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
noncomputable def Equiv [h₁ : IsDirectLimit f g₁] [h₂ : IsDirectLimit f g₂] :
    P₁ ≃ P₂ where
…
```
Mentions: IsDirectLimit
lemma IsDirectLimit.Equiv_apply line 99 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Preorder, LE.le, DirectedSystem, IsDirectLimit, Eq, Equiv, IsDirectLimit.Equiv
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, IsDirectLimit.lift_of, forall_congr, IsDirectLimit.compatibility', eq_self, implies_true
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma IsDirectLimit.linearEquiv_symm_apply line 104 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Preorder, LE.le, DirectedSystem, IsDirectLimit, Eq, Equiv, Equiv.symm, IsDirectLimit.Equiv
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, IsDirectLimit.lift_of, forall_congr, IsDirectLimit.compatibility', eq_self, implies_true
8 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
instance IsDirectLimit.Module.(instance at line 114) line 114 source
theorem IsDirectLimit.Module.compatibility_module line 127 source
no axioms what this means
Statement uses: Preorder, Semiring, AddCommMonoid, Module, LE.le, LinearMap, RingHom.id, DirectedSystem, IsDirectLimit, Eq
Proof uses: IsDirectLimit.compatibility'
8 instances, coercions and auxiliary constants left out.
def IsDirectLimit.Module.lift The universal property of the direct limit: define a linear map from the direct limit by giving a compatible family of linear maps from the components line 135 source
Source documentation
The universal property of the direct limit: define a linear map from the direct limit by giving a compatible family of linear maps from the components. Each element of the limit is sent according to its representative in some component.

standard axioms: Classical.choice, propext what this means
Statement uses: Preorder, Semiring, AddCommMonoid, Module, LE.le, LinearMap, RingHom.id, DirectedSystem, IsDirected, IsDirectLimit, Eq
Proof uses: LinearMap.mk, AddHom.mk, IsDirectLimit.lift
13 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
```
noncomputable def lift
    (Hg : ∀ i j hij x, g₂ j (f i j hij x) = g₂ i x) : P₁ →ₗ[R] P₂ where
…
```
theorem IsDirectLimit.Module.lift_of line 161 source
standard axioms: Classical.choice, propext what this means
Statement uses: Preorder, Semiring, AddCommMonoid, Module, LE.le, LinearMap, RingHom.id, DirectedSystem, IsDirected, IsDirectLimit, Eq, IsDirectLimit.Module.lift
Proof uses: id, IsDirectLimit.lift_of
8 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem IsDirectLimit.Module.lift_unique line 166 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Preorder, Semiring, AddCommMonoid, Module, LE.le, LinearMap, RingHom.id, DirectedSystem, IsDirected, IsDirectLimit, Eq, IsDirectLimit.Module.lift, LinearMap.comp
Proof uses: of_eq_true, Eq.trans, True, congrArg, IsDirectLimit.preimageIndex, IsDirectLimit.preimage, IsDirectLimit.image_preimage, eq_self
12 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
def IsDirectLimit.Module.linearEquiv Any two direct limits of the same directed system are isomorphic line 177 source
Source documentation
Any two direct limits of the same directed system are isomorphic.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Preorder, Semiring, AddCommMonoid, Module, LE.le, LinearMap, RingHom.id, DirectedSystem, IsDirected, IsDirectLimit, LinearEquiv
Proof uses: Equiv, IsDirectLimit.Equiv, LinearEquiv.mk, LinearMap.mk, AddHom.mk, Equiv.toFun, Equiv.invFun, Equiv.left_inv, Equiv.right_inv
15 instances, coercions and auxiliary constants left out.
```
noncomputable def linearEquiv : P₁ ≃ₗ[R] P₂ :=
…
```
lemma IsDirectLimit.Module.linearEquiv_apply line 183 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Preorder, Semiring, AddCommMonoid, Module, LE.le, LinearMap, RingHom.id, DirectedSystem, IsDirected, IsDirectLimit, Eq, LinearEquiv, IsDirectLimit.Module.linearEquiv
Proof uses: IsDirectLimit.Equiv_apply
11 instances, coercions and auxiliary constants left out.
lemma IsDirectLimit.Module.linearEquiv_symm_apply line 188 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Preorder, Semiring, AddCommMonoid, Module, LE.le, LinearMap, RingHom.id, DirectedSystem, IsDirected, IsDirectLimit, Eq, LinearEquiv, LinearEquiv.symm, IsDirectLimit.Module.linearEquiv
Proof uses: IsDirectLimit.linearEquiv_symm_apply
11 instances, coercions and auxiliary constants left out.