Theorem: Zspan.ceil_eq_self_of_mem
File: Mathlib/Algebra/Module/Zlattice/Basic.lean

Proof steps:
state_before: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁴ : NormedLinearOrderedField K
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
b : Basis ι K E
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
h : m ∈ span ℤ (Set.range ⇑b)
⊢ ↑(ceil b m) = m
state_after: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁴ : NormedLinearOrderedField K
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
b : Basis ι K E
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
h : m ∈ span ℤ (Set.range ⇑b)
⊢ ∀ (i : ι), (b.repr ↑(ceil b m)) i = (b.repr m) i
tactic: apply b.ext_elem
state_before: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁴ : NormedLinearOrderedField K
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
b : Basis ι K E
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
h : m ∈ span ℤ (Set.range ⇑b)
⊢ ∀ (i : ι), (b.repr ↑(ceil b m)) i = (b.repr m) i
state_after: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁴ : NormedLinearOrderedField K
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
b : Basis ι K E
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
h : m ∈ span ℤ (Set.range ⇑b)
⊢ ∀ (i : ι), ↑⌈(b.repr m) i⌉ = (b.repr m) i
tactic: simp_rw [repr_ceil_apply b]
state_before: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁴ : NormedLinearOrderedField K
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
b : Basis ι K E
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
h : m ∈ span ℤ (Set.range ⇑b)
⊢ ∀ (i : ι), ↑⌈(b.repr m) i⌉ = (b.repr m) i
state_after: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁴ : NormedLinearOrderedField K
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
b : Basis ι K E
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
h : m ∈ span ℤ (Set.range ⇑b)
i : ι
⊢ ↑⌈(b.repr m) i⌉ = (b.repr m) i
tactic: intro i
state_before: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁴ : NormedLinearOrderedField K
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
b : Basis ι K E
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
h : m ∈ span ℤ (Set.range ⇑b)
i : ι
⊢ ↑⌈(b.repr m) i⌉ = (b.repr m) i
state_after: case intro
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁴ : NormedLinearOrderedField K
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
b : Basis ι K E
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
h : m ∈ span ℤ (Set.range ⇑b)
i : ι
z : ℤ
hz : (algebraMap ℤ K) z = (b.repr m) i
⊢ ↑⌈(b.repr m) i⌉ = (b.repr m) i
tactic: obtain ⟨z, hz⟩ := (b.mem_span_iff_repr_mem ℤ _).mp h i
state_before: case intro
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁴ : NormedLinearOrderedField K
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
b : Basis ι K E
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
h : m ∈ span ℤ (Set.range ⇑b)
i : ι
z : ℤ
hz : (algebraMap ℤ K) z = (b.repr m) i
⊢ ↑⌈(b.repr m) i⌉ = (b.repr m) i
state_after: case intro
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁴ : NormedLinearOrderedField K
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
b : Basis ι K E
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
h : m ∈ span ℤ (Set.range ⇑b)
i : ι
z : ℤ
hz : (algebraMap ℤ K) z = (b.repr m) i
⊢ ↑⌈(algebraMap ℤ K) z⌉ = (algebraMap ℤ K) z
tactic: rw [← hz]
state_before: case intro
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁴ : NormedLinearOrderedField K
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
b : Basis ι K E
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
h : m ∈ span ℤ (Set.range ⇑b)
i : ι
z : ℤ
hz : (algebraMap ℤ K) z = (b.repr m) i
⊢ ↑⌈(algebraMap ℤ K) z⌉ = (algebraMap ℤ K) z
state_after: no goals
tactic: exact congr_arg (Int.cast : ℤ → K) (Int.ceil_intCast z)