Theorem: Zlattice.FG
File: Mathlib/Algebra/Module/Zlattice/Basic.lean

Proof steps:
state_before: K : Type u_1
inst✝⁷ : NormedLinearOrderedField K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : AddSubgroup E
inst✝ : DiscreteTopology ↥L
hs : IsZlattice K L
⊢ L.FG
state_after: K : Type u_1
inst✝⁷ : NormedLinearOrderedField K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : AddSubgroup E
inst✝ : DiscreteTopology ↥L
hs : IsZlattice K L
⊢ (AddSubgroup.toIntSubmodule L).FG
tactic: suffices (AddSubgroup.toIntSubmodule L).FG by exact (fg_iff_add_subgroup_fg _).mp this
state_before: K : Type u_1
inst✝⁷ : NormedLinearOrderedField K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : AddSubgroup E
inst✝ : DiscreteTopology ↥L
hs : IsZlattice K L
⊢ (AddSubgroup.toIntSubmodule L).FG
state_after: case intro.intro.intro
K : Type u_1
inst✝⁷ : NormedLinearOrderedField K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : AddSubgroup E
inst✝ : DiscreteTopology ↥L
hs : IsZlattice K L
s : Set E
h_incl : s ⊆ ↑L
h_span : span K s = span K ↑L
h_lind : LinearIndependent (ι := { x // x ∈ s }) K Subtype.val
⊢ (AddSubgroup.toIntSubmodule L).FG
tactic: obtain ⟨s, ⟨h_incl, ⟨h_span, h_lind⟩⟩⟩ := exists_linearIndependent K (L : Set E)
state_before: case intro.intro.intro
K : Type u_1
inst✝⁷ : NormedLinearOrderedField K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : AddSubgroup E
inst✝ : DiscreteTopology ↥L
hs : IsZlattice K L
s : Set E
h_incl : s ⊆ ↑L
h_span : span K s = span K ↑L
h_lind : LinearIndependent (ι := { x // x ∈ s }) K Subtype.val
⊢ (AddSubgroup.toIntSubmodule L).FG
state_after: case intro.intro.intro.refine_1
K : Type u_1
inst✝⁷ : NormedLinearOrderedField K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : AddSubgroup E
inst✝ : DiscreteTopology ↥L
hs : IsZlattice K L
s : Set E
h_incl : s ⊆ ↑L
h_span : span K s = span K ↑L
h_lind : LinearIndependent (ι := { x // x ∈ s }) K Subtype.val
⊢ (map (span ℤ s).mkQ (AddSubgroup.toIntSubmodule L)).FG

case intro.intro.intro.refine_2
K : Type u_1
inst✝⁷ : NormedLinearOrderedField K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : AddSubgroup E
inst✝ : DiscreteTopology ↥L
hs : IsZlattice K L
s : Set E
h_incl : s ⊆ ↑L
h_span : span K s = span K ↑L
h_lind : LinearIndependent (ι := { x // x ∈ s }) K Subtype.val
⊢ (AddSubgroup.toIntSubmodule L ⊓ LinearMap.ker (span ℤ s).mkQ).FG
tactic: refine fg_of_fg_map_of_fg_inf_ker (span ℤ s).mkQ ?_ ?_