Theorem: Pi.linearIndependent_stdBasis
File: Mathlib/LinearAlgebra/StdBasis.lean

Proof steps:
state_before: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
⊢ LinearIndependent R fun ji => (stdBasis R Ms ji.fst) (v ji.fst ji.snd)
state_after: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)
⊢ LinearIndependent R fun ji => (stdBasis R Ms ji.fst) (v ji.fst ji.snd)
tactic: have hs' : ∀ j : η, LinearIndependent R fun i : ιs j => stdBasis R Ms j (v j i) := by
    intro j
    exact (hs j).map' _ (ker_stdBasis _ _ _)
state_before: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)
⊢ LinearIndependent R fun ji => (stdBasis R Ms ji.fst) (v ji.fst ji.snd)
state_after: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)
⊢ ∀ (i : η) (t : Set η),
    t.Finite →
      i ∉ t →
        Disjoint (span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))
          (⨆ i ∈ t, span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))
tactic: apply linearIndependent_iUnion_finite hs'
state_before: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)
⊢ ∀ (i : η) (t : Set η),
    t.Finite →
      i ∉ t →
        Disjoint (span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))
          (⨆ i ∈ t, span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))
state_after: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)
j : η
J : Set η
a✝ : J.Finite
hiJ : j ∉ J
⊢ Disjoint (span R (Set.range fun i => (stdBasis R Ms j) (v j i)))
    (⨆ i ∈ J, span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))
tactic: intro j J _ hiJ
state_before: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)
j : η
J : Set η
a✝ : J.Finite
hiJ : j ∉ J
⊢ Disjoint (span R (Set.range fun i => (stdBasis R Ms j) (v j i)))
    (⨆ i ∈ J, span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))
state_after: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)
j : η
J : Set η
a✝ : J.Finite
hiJ : j ∉ J
h₀ : ∀ (j : η), span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)
⊢ Disjoint (span R (Set.range fun i => (stdBasis R Ms j) (v j i)))
    (⨆ i ∈ J, span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))
tactic: have h₀ :
    ∀ j, span R (range fun i : ιs j => stdBasis R Ms j (v j i)) ≤
      LinearMap.range (stdBasis R Ms j) := by
    intro j
    rw [span_le, LinearMap.range_coe]
    apply range_comp_subset_range
state_before: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)
j : η
J : Set η
a✝ : J.Finite
hiJ : j ∉ J
h₀ : ∀ (j : η), span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)
⊢ Disjoint (span R (Set.range fun i => (stdBasis R Ms j) (v j i)))
    (⨆ i ∈ J, span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))
state_after: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)
j : η
J : Set η
a✝ : J.Finite
hiJ : j ∉ J
h₀ : ∀ (j : η), span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)
h₁ : span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤ ⨆ i ∈ {j}, LinearMap.range (stdBasis R Ms i)
⊢ Disjoint (span R (Set.range fun i => (stdBasis R Ms j) (v j i)))
    (⨆ i ∈ J, span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))
tactic: have h₁ :
    span R (range fun i : ιs j => stdBasis R Ms j (v j i)) ≤
      ⨆ i ∈ ({j} : Set _), LinearMap.range (stdBasis R Ms i) := by
    rw [@iSup_singleton _ _ _ fun i => LinearMap.range (stdBasis R (Ms) i)]
    apply h₀
state_before: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)
j : η
J : Set η
a✝ : J.Finite
hiJ : j ∉ J
h₀ : ∀ (j : η), span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)
h₁ : span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤ ⨆ i ∈ {j}, LinearMap.range (stdBasis R Ms i)
⊢ Disjoint (span R (Set.range fun i => (stdBasis R Ms j) (v j i)))
    (⨆ i ∈ J, span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))
state_after: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)
j : η
J : Set η
a✝ : J.Finite
hiJ : j ∉ J
h₀ : ∀ (j : η), span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)
h₁ : span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤ ⨆ i ∈ {j}, LinearMap.range (stdBasis R Ms i)
h₂ :
  ⨆ j ∈ J, span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤
    ⨆ j ∈ J, LinearMap.range (stdBasis R (fun j => Ms j) j)
⊢ Disjoint (span R (Set.range fun i => (stdBasis R Ms j) (v j i)))
    (⨆ i ∈ J, span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))
tactic: have h₂ :
    ⨆ j ∈ J, span R (range fun i : ιs j => stdBasis R Ms j (v j i)) ≤
      ⨆ j ∈ J, LinearMap.range (stdBasis R (fun j : η => Ms j) j) :=
    iSup₂_mono fun i _ => h₀ i
state_before: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)
j : η
J : Set η
a✝ : J.Finite
hiJ : j ∉ J
h₀ : ∀ (j : η), span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)
h₁ : span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤ ⨆ i ∈ {j}, LinearMap.range (stdBasis R Ms i)
h₂ :
  ⨆ j ∈ J, span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤
    ⨆ j ∈ J, LinearMap.range (stdBasis R (fun j => Ms j) j)
⊢ Disjoint (span R (Set.range fun i => (stdBasis R Ms j) (v j i)))
    (⨆ i ∈ J, span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))
state_after: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)
j : η
J : Set η
a✝ : J.Finite
hiJ : j ∉ J
h₀ : ∀ (j : η), span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)
h₁ : span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤ ⨆ i ∈ {j}, LinearMap.range (stdBasis R Ms i)
h₂ :
  ⨆ j ∈ J, span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤
    ⨆ j ∈ J, LinearMap.range (stdBasis R (fun j => Ms j) j)
h₃ : Disjoint (fun i => i ∈ {j}) J
⊢ Disjoint (span R (Set.range fun i => (stdBasis R Ms j) (v j i)))
    (⨆ i ∈ J, span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))
tactic: have h₃ : Disjoint (fun i : η => i ∈ ({j} : Set _)) J := by
    convert Set.disjoint_singleton_left.2 hiJ using 0
state_before: R : Type u_1
η : Type u_2
ιs : η → Type u_3
Ms : η → Type u_4
inst✝³ : Ring R
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module R (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent R (v i)
hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)
j : η
J : Set η
a✝ : J.Finite
hiJ : j ∉ J
h₀ : ∀ (j : η), span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)
h₁ : span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤ ⨆ i ∈ {j}, LinearMap.range (stdBasis R Ms i)
h₂ :
  ⨆ j ∈ J, span R (Set.range fun i => (stdBasis R Ms j) (v j i)) ≤
    ⨆ j ∈ J, LinearMap.range (stdBasis R (fun j => Ms j) j)
h₃ : Disjoint (fun i => i ∈ {j}) J
⊢ Disjoint (span R (Set.range fun i => (stdBasis R Ms j) (v j i)))
    (⨆ i ∈ J, span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))
state_after: no goals
tactic: exact (disjoint_stdBasis_stdBasis _ _ _ _ h₃).mono h₁ h₂