ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- rank_add_rank_split
theorem rank_add_rank_split (db : V₂ →ₗ[K] V) (eb : V₃ →ₗ[K] V) (cd : V₁ →ₗ[K] V₂)
    (ce : V₁ →ₗ[K] V₃) (hde : ⊤ ≤ LinearMap.range db ⊔ LinearMap.range eb) (hgd : ker cd = ⊥)
    (eq : db.comp cd = eb.comp ce) (eq₂ : ∀ d e, db d = eb e → ∃ c, cd c = d ∧ ce c = e) :
    Module.rank K V + Module.rank K V₁ = Module.rank K V₂ + Module.rank K V₃ 
Arguments: {K : Type u} {V V₁ V₂ V₃ : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] [AddCommGroup V₃] [Module K V₃] (db : V₂ →ₗ[K] V) (eb : V₃ →ₗ[K] V) (cd : V₁ →ₗ[K] V₂) (ce : V₁ →ₗ[K] V₃) (hde : ⊤ ≤ LinearMap.range db ⊔ LinearMap.range eb) (hgd : LinearMap.ker cd = ⊥) (eq : db ∘ₗ cd = eb ∘ₗ ce) (eq₂ : ∀ (d : V₂) (e : V₃), db d = eb e → ∃ c, cd c = d ∧ ce c = e)
Goal: Module.rank K V + Module.rank K V₁ = Module.rank K V₂ + Module.rank K V₃
-- linearIndependent_of_top_le_span_of_card_eq_finrank
theorem linearIndependent_of_top_le_span_of_card_eq_finrank {ι : Type*} [Fintype ι] {b : ι → V}
    (spans : ⊤ ≤ span K (Set.range b)) (card_eq : Fintype.card ι = finrank K V) :
    LinearIndependent K b 
Arguments: {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type u_2} [Fintype ι] {b : ι → V} (spans : ⊤ ≤ Submodule.span K (Set.range b)) (card_eq : Fintype.card ι = FiniteDimensional.finrank K V)
Goal: LinearIndependent K b
-- linearIndependent_iff_card_eq_finrank_span
theorem linearIndependent_iff_card_eq_finrank_span {ι : Type*} [Fintype ι] {b : ι → V} :
    LinearIndependent K b ↔ Fintype.card ι = (Set.range b).finrank K 
Arguments: {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type u_2} [Fintype ι] {b : ι → V}
Goal: LinearIndependent K b ↔ Fintype.card ι = Set.finrank K (Set.range b)
-- linearIndependent_iff_card_le_finrank_span
theorem linearIndependent_iff_card_le_finrank_span {ι : Type*} [Fintype ι] {b : ι → V} :
    LinearIndependent K b ↔ Fintype.card ι ≤ (Set.range b).finrank K 
Arguments: {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type u_2} [Fintype ι] {b : ι → V}
Goal: LinearIndependent K b ↔ Fintype.card ι ≤ Set.finrank K (Set.range b)
-- coe_basisOfTopLeSpanOfCardEqFinrank
@[simp]
theorem coe_basisOfTopLeSpanOfCardEqFinrank {ι : Type*} [Fintype ι] (b : ι → V)
    (le_span : ⊤ ≤ span K (Set.range b)) (card_eq : Fintype.card ι = finrank K V) :
    ⇑(basisOfTopLeSpanOfCardEqFinrank b le_span card_eq) = b 
Arguments: {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type u_2} [Fintype ι] (b : ι → V) (le_span : ⊤ ≤ Submodule.span K (Set.range b)) (card_eq : Fintype.card ι = FiniteDimensional.finrank K V)
Goal: ⇑(basisOfTopLeSpanOfCardEqFinrank b le_span card_eq) = b
-- finsetBasisOfTopLeSpanOfCardEqFinrank_repr_apply
@[simps! repr_apply]
noncomputable def finsetBasisOfTopLeSpanOfCardEqFinrank {s : Finset V}
    (le_span : ⊤ ≤ span K (s : Set V)) (card_eq : s.card = finrank K V) : Basis {x // x ∈ s} K V 
Arguments: {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Finset V} (le_span : ⊤ ≤ Submodule.span K ↑s) (card_eq : s.card = FiniteDimensional.finrank K V)
Goal: ∀ (a : V),
  (finsetBasisOfTopLeSpanOfCardEqFinrank le_span card_eq).repr a =
    ⋯.repr ((LinearMap.codRestrict (Submodule.span K (Set.range Subtype.val)) LinearMap.id ⋯) a)
-- setBasisOfTopLeSpanOfCardEqFinrank_repr_apply
@[simps! repr_apply]
noncomputable def setBasisOfTopLeSpanOfCardEqFinrank {s : Set V} [Fintype s]
    (le_span : ⊤ ≤ span K s) (card_eq : s.toFinset.card = finrank K V) : Basis s K V 
Arguments: {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} [Fintype ↑s] (le_span : ⊤ ≤ Submodule.span K s) (card_eq : s.toFinset.card = FiniteDimensional.finrank K V)
Goal: ∀ (a : V),
  (setBasisOfTopLeSpanOfCardEqFinrank le_span card_eq).repr a =
    ⋯.repr ((LinearMap.codRestrict (Submodule.span K (Set.range Subtype.val)) LinearMap.id ⋯) a)
-- max_aleph0_card_le_rank_fun_nat
theorem max_aleph0_card_le_rank_fun_nat : max ℵ₀ #K ≤ Module.rank K (ℕ → K) 
Arguments: (K : Type u) [DivisionRing K]
Goal: max ℵ₀ #K ≤ Module.rank K (ℕ → K)
-- rank_fun_infinite
theorem rank_fun_infinite {ι : Type v} [hι : Infinite ι] : Module.rank K (ι → K) = #(ι → K) 
Arguments: {K : Type u} [DivisionRing K] {ι : Type v} [hι : Infinite ι]
Goal: Module.rank K (ι → K) = #(ι → K)
-- rank_dual_eq_card_dual_of_aleph0_le_rank'
theorem rank_dual_eq_card_dual_of_aleph0_le_rank' {V : Type*} [AddCommGroup V] [Module K V]
    (h : ℵ₀ ≤ Module.rank K V) : Module.rank Kᵐᵒᵖ (V →ₗ[K] K) = #(V →ₗ[K] K) 
Arguments: {K : Type u} [DivisionRing K] {V : Type u_2} [AddCommGroup V] [Module K V] (h : ℵ₀ ≤ Module.rank K V)
Goal: Module.rank Kᵐᵒᵖ (V →ₗ[K] K) = #(V →ₗ[K] K)
-- rank_dual_eq_card_dual_of_aleph0_le_rank
theorem rank_dual_eq_card_dual_of_aleph0_le_rank {K V} [Field K] [AddCommGroup V] [Module K V]
    (h : ℵ₀ ≤ Module.rank K V) : Module.rank K (V →ₗ[K] K) = #(V →ₗ[K] K) 
Arguments: {K : Type u_2} {V : Type u_3} [Field K] [AddCommGroup V] [Module K V] (h : ℵ₀ ≤ Module.rank K V)
Goal: Module.rank K (V →ₗ[K] K) = #(V →ₗ[K] K)
-- lift_rank_lt_rank_dual'
theorem lift_rank_lt_rank_dual' {V : Type v} [AddCommGroup V] [Module K V]
    (h : ℵ₀ ≤ Module.rank K V) :
    Cardinal.lift.{u} (Module.rank K V) < Module.rank Kᵐᵒᵖ (V →ₗ[K] K) 
Arguments: {K : Type u} [DivisionRing K] {V : Type v} [AddCommGroup V] [Module K V] (h : ℵ₀ ≤ Module.rank K V)
Goal: Cardinal.lift.{u, v} (Module.rank K V) < Module.rank Kᵐᵒᵖ (V →ₗ[K] K)
-- lift_rank_lt_rank_dual
theorem lift_rank_lt_rank_dual {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V]
    (h : ℵ₀ ≤ Module.rank K V) :
    Cardinal.lift.{u} (Module.rank K V) < Module.rank K (V →ₗ[K] K) 
Arguments: {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (h : ℵ₀ ≤ Module.rank K V)
Goal: Cardinal.lift.{u, v} (Module.rank K V) < Module.rank K (V →ₗ[K] K)
-- rank_lt_rank_dual'
theorem rank_lt_rank_dual' {V : Type u} [AddCommGroup V] [Module K V] (h : ℵ₀ ≤ Module.rank K V) :
    Module.rank K V < Module.rank Kᵐᵒᵖ (V →ₗ[K] K) 
Arguments: {K : Type u} [DivisionRing K] {V : Type u} [AddCommGroup V] [Module K V] (h : ℵ₀ ≤ Module.rank K V)
Goal: Module.rank K V < Module.rank Kᵐᵒᵖ (V →ₗ[K] K)
-- rank_lt_rank_dual
theorem rank_lt_rank_dual {K V : Type u} [Field K] [AddCommGroup V] [Module K V]
    (h : ℵ₀ ≤ Module.rank K V) : Module.rank K V < Module.rank K (V →ₗ[K] K) 
Arguments: {K V : Type u} [Field K] [AddCommGroup V] [Module K V] (h : ℵ₀ ≤ Module.rank K V)
Goal: Module.rank K V < Module.rank K (V →ₗ[K] K)
-- TensorProduct.finsuppLeft_apply_tmul
lemma finsuppLeft_apply_tmul (p : ι →₀ M) (n : N) :
    finsuppLeft R M N ι (p ⊗ₜ[R] n) = p.sum fun i m ↦ Finsupp.single i (m ⊗ₜ[R] n) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_4} [DecidableEq ι] (p : ι →₀ M) (n : N)
Goal: (TensorProduct.finsuppLeft R M N ι) (p ⊗ₜ[R] n) = p.sum fun i m => Finsupp.single i (m ⊗ₜ[R] n)
-- TensorProduct.finsuppLeft_apply_tmul_apply
@[simp]
lemma finsuppLeft_apply_tmul_apply (p : ι →₀ M) (n : N) (i : ι) :
    finsuppLeft R M N ι (p ⊗ₜ[R] n) i = p i ⊗ₜ[R] n 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_4} [DecidableEq ι] (p : ι →₀ M) (n : N) (i : ι)
Goal: ((TensorProduct.finsuppLeft R M N ι) (p ⊗ₜ[R] n)) i = p i ⊗ₜ[R] n
-- TensorProduct.finsuppLeft_apply
theorem finsuppLeft_apply (t : (ι →₀ M) ⊗[R] N) (i : ι) :
    finsuppLeft R M N ι t i = rTensor N (Finsupp.lapply i) t 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_4} [DecidableEq ι] (t : (ι →₀ M) ⊗[R] N) (i : ι)
Goal: ((TensorProduct.finsuppLeft R M N ι) t) i = (LinearMap.rTensor N (Finsupp.lapply i)) t
-- TensorProduct.finsuppLeft_symm_apply_single
@[simp]
lemma finsuppLeft_symm_apply_single (i : ι) (m : M) (n : N) :
    (finsuppLeft R M N ι).symm (Finsupp.single i (m ⊗ₜ[R] n)) =
      Finsupp.single i m ⊗ₜ[R] n 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_4} [DecidableEq ι] (i : ι) (m : M) (n : N)
Goal: (TensorProduct.finsuppLeft R M N ι).symm (Finsupp.single i (m ⊗ₜ[R] n)) = Finsupp.single i m ⊗ₜ[R] n
-- TensorProduct.finsuppRight_apply_tmul
lemma finsuppRight_apply_tmul (m : M) (p : ι →₀ N) :
    finsuppRight R M N ι (m ⊗ₜ[R] p) = p.sum fun i n ↦ Finsupp.single i (m ⊗ₜ[R] n) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_4} [DecidableEq ι] (m : M) (p : ι →₀ N)
Goal: (TensorProduct.finsuppRight R M N ι) (m ⊗ₜ[R] p) = p.sum fun i n => Finsupp.single i (m ⊗ₜ[R] n)
-- TensorProduct.finsuppRight_apply_tmul_apply
@[simp]
lemma finsuppRight_apply_tmul_apply (m : M) (p : ι →₀ N) (i : ι) :
    finsuppRight R M N ι (m ⊗ₜ[R] p) i = m ⊗ₜ[R] p i 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_4} [DecidableEq ι] (m : M) (p : ι →₀ N) (i : ι)
Goal: ((TensorProduct.finsuppRight R M N ι) (m ⊗ₜ[R] p)) i = m ⊗ₜ[R] p i
-- TensorProduct.finsuppRight_apply
theorem finsuppRight_apply (t : M ⊗[R] (ι →₀ N)) (i : ι) :
    finsuppRight R M N ι t i = lTensor M (Finsupp.lapply i) t 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_4} [DecidableEq ι] (t : M ⊗[R] (ι →₀ N)) (i : ι)
Goal: ((TensorProduct.finsuppRight R M N ι) t) i = (LinearMap.lTensor M (Finsupp.lapply i)) t
-- TensorProduct.finsuppRight_symm_apply_single
@[simp]
lemma finsuppRight_symm_apply_single (i : ι) (m : M) (n : N) :
    (finsuppRight R M N ι).symm (Finsupp.single i (m ⊗ₜ[R] n)) =
      m ⊗ₜ[R] Finsupp.single i n 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_4} [DecidableEq ι] (i : ι) (m : M) (n : N)
Goal: (TensorProduct.finsuppRight R M N ι).symm (Finsupp.single i (m ⊗ₜ[R] n)) = m ⊗ₜ[R] Finsupp.single i n
-- TensorProduct.finsuppLeft_smul'
lemma finsuppLeft_smul' (s : S) (t : (ι →₀ M) ⊗[R] N) :
    finsuppLeft R M N ι (s • t) = s • finsuppLeft R M N ι t 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_4} [DecidableEq ι] {S : Type u_5} [CommSemiring S] [Algebra R S] [Module S M] [IsScalarTower R S M] (s : S) (t : (ι →₀ M) ⊗[R] N)
Goal: (TensorProduct.finsuppLeft R M N ι) (s • t) = s • (TensorProduct.finsuppLeft R M N ι) t
-- TensorProduct.finsuppLeft'_apply
lemma finsuppLeft'_apply (x : (ι →₀ M) ⊗[R] N) :
    finsuppLeft' R M N ι S x = finsuppLeft R M N ι x 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_4} [DecidableEq ι] {S : Type u_5} [CommSemiring S] [Algebra R S] [Module S M] [IsScalarTower R S M] (x : (ι →₀ M) ⊗[R] N)
Goal: (TensorProduct.finsuppLeft' R M N ι S) x = (TensorProduct.finsuppLeft R M N ι) x
-- TensorProduct.finsuppScalarLeft_apply_tmul_apply
@[simp]
lemma finsuppScalarLeft_apply_tmul_apply (p : ι →₀ R) (n : N) (i : ι) :
    finsuppScalarLeft R N ι (p ⊗ₜ[R] n) i = p i • n 
Arguments: {R : Type u_1} [CommSemiring R] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_4} [DecidableEq ι] (p : ι →₀ R) (n : N) (i : ι)
Goal: ((TensorProduct.finsuppScalarLeft R N ι) (p ⊗ₜ[R] n)) i = p i • n
-- TensorProduct.finsuppScalarLeft_apply_tmul
lemma finsuppScalarLeft_apply_tmul (p : ι →₀ R) (n : N) :
    finsuppScalarLeft R N ι (p ⊗ₜ[R] n) = p.sum fun i m ↦ Finsupp.single i (m • n) 
Arguments: {R : Type u_1} [CommSemiring R] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_4} [DecidableEq ι] (p : ι →₀ R) (n : N)
Goal: (TensorProduct.finsuppScalarLeft R N ι) (p ⊗ₜ[R] n) = p.sum fun i m => Finsupp.single i (m • n)
-- TensorProduct.finsuppScalarLeft_apply
lemma finsuppScalarLeft_apply (pn : (ι →₀ R) ⊗[R] N) (i : ι) :
    finsuppScalarLeft R N ι pn i = TensorProduct.lid R N ((Finsupp.lapply i).rTensor N pn) 
Arguments: {R : Type u_1} [CommSemiring R] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_4} [DecidableEq ι] (pn : (ι →₀ R) ⊗[R] N) (i : ι)
Goal: ((TensorProduct.finsuppScalarLeft R N ι) pn) i = (TensorProduct.lid R N) ((LinearMap.rTensor N (Finsupp.lapply i)) pn)
-- TensorProduct.finsuppScalarLeft_symm_apply_single
@[simp]
lemma finsuppScalarLeft_symm_apply_single (i : ι) (n : N) :
    (finsuppScalarLeft R N ι).symm (Finsupp.single i n) =
      (Finsupp.single i 1) ⊗ₜ[R] n 
Arguments: {R : Type u_1} [CommSemiring R] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_4} [DecidableEq ι] (i : ι) (n : N)
Goal: (TensorProduct.finsuppScalarLeft R N ι).symm (Finsupp.single i n) = Finsupp.single i 1 ⊗ₜ[R] n
-- TensorProduct.finsuppScalarRight_apply_tmul_apply
@[simp]
lemma finsuppScalarRight_apply_tmul_apply (m : M) (p : ι →₀ R) (i : ι) :
    finsuppScalarRight R M ι (m ⊗ₜ[R] p) i = p i • m 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {ι : Type u_4} [DecidableEq ι] (m : M) (p : ι →₀ R) (i : ι)
Goal: ((TensorProduct.finsuppScalarRight R M ι) (m ⊗ₜ[R] p)) i = p i • m
-- TensorProduct.finsuppScalarRight_apply_tmul
lemma finsuppScalarRight_apply_tmul (m : M) (p : ι →₀ R) :
    finsuppScalarRight R M ι (m ⊗ₜ[R] p) = p.sum fun i n ↦ Finsupp.single i (n • m) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {ι : Type u_4} [DecidableEq ι] (m : M) (p : ι →₀ R)
Goal: (TensorProduct.finsuppScalarRight R M ι) (m ⊗ₜ[R] p) = p.sum fun i n => Finsupp.single i (n • m)
-- TensorProduct.finsuppScalarRight_apply
lemma finsuppScalarRight_apply (t : M ⊗[R] (ι →₀ R)) (i : ι) :
    finsuppScalarRight R M ι t i = TensorProduct.rid R M ((Finsupp.lapply i).lTensor M t) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {ι : Type u_4} [DecidableEq ι] (t : M ⊗[R] (ι →₀ R)) (i : ι)
Goal: ((TensorProduct.finsuppScalarRight R M ι) t) i = (TensorProduct.rid R M) ((LinearMap.lTensor M (Finsupp.lapply i)) t)
-- TensorProduct.finsuppScalarRight_symm_apply_single
@[simp]
lemma finsuppScalarRight_symm_apply_single (i : ι) (m : M) :
    (finsuppScalarRight R M ι).symm (Finsupp.single i m) =
      m ⊗ₜ[R] (Finsupp.single i 1) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {ι : Type u_4} [DecidableEq ι] (i : ι) (m : M)
Goal: (TensorProduct.finsuppScalarRight R M ι).symm (Finsupp.single i m) = m ⊗ₜ[R] Finsupp.single i 1
-- finsuppTensorFinsupp_single
@[simp]
theorem finsuppTensorFinsupp_single (i : ι) (m : M) (k : κ) (n : N) :
    finsuppTensorFinsupp R S M N ι κ (Finsupp.single i m ⊗ₜ Finsupp.single k n) =
      Finsupp.single (i, k) (m ⊗ₜ n) 
Arguments: (R : Type u_1) (S : Type u_2) (M : Type u_3) (N : Type u_4) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Semiring S] [Algebra R S] [Module S M] [IsScalarTower R S M] (i : ι) (m : M) (k : κ) (n : N)
Goal: (finsuppTensorFinsupp R S M N ι κ) (Finsupp.single i m ⊗ₜ[R] Finsupp.single k n) = Finsupp.single (i, k) (m ⊗ₜ[R] n)
-- finsuppTensorFinsupp_apply
@[simp]
theorem finsuppTensorFinsupp_apply (f : ι →₀ M) (g : κ →₀ N) (i : ι) (k : κ) :
    finsuppTensorFinsupp R S M N ι κ (f ⊗ₜ g) (i, k) = f i ⊗ₜ g k 
Arguments: (R : Type u_1) (S : Type u_2) (M : Type u_3) (N : Type u_4) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Semiring S] [Algebra R S] [Module S M] [IsScalarTower R S M] (f : ι →₀ M) (g : κ →₀ N) (i : ι) (k : κ)
Goal: ((finsuppTensorFinsupp R S M N ι κ) (f ⊗ₜ[R] g)) (i, k) = f i ⊗ₜ[R] g k
-- finsuppTensorFinsupp_symm_single
@[simp]
theorem finsuppTensorFinsupp_symm_single (i : ι × κ) (m : M) (n : N) :
    (finsuppTensorFinsupp R S M N ι κ).symm (Finsupp.single i (m ⊗ₜ n)) =
      Finsupp.single i.1 m ⊗ₜ Finsupp.single i.2 n 
Arguments: (R : Type u_1) (S : Type u_2) (M : Type u_3) (N : Type u_4) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Semiring S] [Algebra R S] [Module S M] [IsScalarTower R S M] (i : ι × κ) (m : M) (n : N)
Goal: (finsuppTensorFinsupp R S M N ι κ).symm (Finsupp.single i (m ⊗ₜ[R] n)) = Finsupp.single i.1 m ⊗ₜ[R] Finsupp.single i.2 n
-- finsuppTensorFinsuppLid_apply_apply
@[simp]
theorem finsuppTensorFinsuppLid_apply_apply (f : ι →₀ R) (g : κ →₀ N) (a : ι) (b : κ) :
    finsuppTensorFinsuppLid R N ι κ (f ⊗ₜ[R] g) (a, b) = f a • g b 
Arguments: (R : Type u_1) (N : Type u_4) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid N] [Module R N] (f : ι →₀ R) (g : κ →₀ N) (a : ι) (b : κ)
Goal: ((finsuppTensorFinsuppLid R N ι κ) (f ⊗ₜ[R] g)) (a, b) = f a • g b
-- finsuppTensorFinsuppLid_single_tmul_single
@[simp]
theorem finsuppTensorFinsuppLid_single_tmul_single (a : ι) (b : κ) (r : R) (n : N) :
    finsuppTensorFinsuppLid R N ι κ (Finsupp.single a r ⊗ₜ[R] Finsupp.single b n) =
      Finsupp.single (a, b) (r • n) 
Arguments: (R : Type u_1) (N : Type u_4) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid N] [Module R N] (a : ι) (b : κ) (r : R) (n : N)
Goal: (finsuppTensorFinsuppLid R N ι κ) (Finsupp.single a r ⊗ₜ[R] Finsupp.single b n) = Finsupp.single (a, b) (r • n)
-- finsuppTensorFinsuppLid_symm_single_smul
@[simp]
theorem finsuppTensorFinsuppLid_symm_single_smul (i : ι × κ) (r : R) (n : N) :
    (finsuppTensorFinsuppLid R N ι κ).symm (Finsupp.single i (r • n)) =
      Finsupp.single i.1 r ⊗ₜ Finsupp.single i.2 n 
Arguments: (R : Type u_1) (N : Type u_4) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid N] [Module R N] (i : ι × κ) (r : R) (n : N)
Goal: (finsuppTensorFinsuppLid R N ι κ).symm (Finsupp.single i (r • n)) = Finsupp.single i.1 r ⊗ₜ[R] Finsupp.single i.2 n
-- finsuppTensorFinsuppRid_apply_apply
@[simp]
theorem finsuppTensorFinsuppRid_apply_apply (f : ι →₀ M) (g : κ →₀ R) (a : ι) (b : κ) :
    finsuppTensorFinsuppRid R M ι κ (f ⊗ₜ[R] g) (a, b) = g b • f a 
Arguments: (R : Type u_1) (M : Type u_3) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid M] [Module R M] (f : ι →₀ M) (g : κ →₀ R) (a : ι) (b : κ)
Goal: ((finsuppTensorFinsuppRid R M ι κ) (f ⊗ₜ[R] g)) (a, b) = g b • f a
-- finsuppTensorFinsuppRid_single_tmul_single
@[simp]
theorem finsuppTensorFinsuppRid_single_tmul_single (a : ι) (b : κ) (m : M) (r : R) :
    finsuppTensorFinsuppRid R M ι κ (Finsupp.single a m ⊗ₜ[R] Finsupp.single b r) =
      Finsupp.single (a, b) (r • m) 
Arguments: (R : Type u_1) (M : Type u_3) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid M] [Module R M] (a : ι) (b : κ) (m : M) (r : R)
Goal: (finsuppTensorFinsuppRid R M ι κ) (Finsupp.single a m ⊗ₜ[R] Finsupp.single b r) = Finsupp.single (a, b) (r • m)
-- finsuppTensorFinsuppRid_symm_single_smul
@[simp]
theorem finsuppTensorFinsuppRid_symm_single_smul (i : ι × κ) (m : M) (r : R) :
    (finsuppTensorFinsuppRid R M ι κ).symm (Finsupp.single i (r • m)) =
      Finsupp.single i.1 m ⊗ₜ Finsupp.single i.2 r 
Arguments: (R : Type u_1) (M : Type u_3) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid M] [Module R M] (i : ι × κ) (m : M) (r : R)
Goal: (finsuppTensorFinsuppRid R M ι κ).symm (Finsupp.single i (r • m)) = Finsupp.single i.1 m ⊗ₜ[R] Finsupp.single i.2 r
-- finsuppTensorFinsupp'_apply_apply
@[simp]
theorem finsuppTensorFinsupp'_apply_apply (f : ι →₀ R) (g : κ →₀ R) (a : ι) (b : κ) :
    finsuppTensorFinsupp' R ι κ (f ⊗ₜ[R] g) (a, b) = f a * g b 
Arguments: (R : Type u_1) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] (f : ι →₀ R) (g : κ →₀ R) (a : ι) (b : κ)
Goal: ((finsuppTensorFinsupp' R ι κ) (f ⊗ₜ[R] g)) (a, b) = f a * g b
-- finsuppTensorFinsupp'_single_tmul_single
@[simp]
theorem finsuppTensorFinsupp'_single_tmul_single (a : ι) (b : κ) (r₁ r₂ : R) :
    finsuppTensorFinsupp' R ι κ (Finsupp.single a r₁ ⊗ₜ[R] Finsupp.single b r₂) =
      Finsupp.single (a, b) (r₁ * r₂) 
Arguments: (R : Type u_1) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] (a : ι) (b : κ) (r₁ r₂ : R)
Goal: (finsuppTensorFinsupp' R ι κ) (Finsupp.single a r₁ ⊗ₜ[R] Finsupp.single b r₂) = Finsupp.single (a, b) (r₁ * r₂)
-- finsuppTensorFinsupp'_symm_single_mul
theorem finsuppTensorFinsupp'_symm_single_mul (i : ι × κ) (r₁ r₂ : R) :
    (finsuppTensorFinsupp' R ι κ).symm (Finsupp.single i (r₁ * r₂)) =
      Finsupp.single i.1 r₁ ⊗ₜ Finsupp.single i.2 r₂ 
Arguments: (R : Type u_1) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] (i : ι × κ) (r₁ r₂ : R)
Goal: (finsuppTensorFinsupp' R ι κ).symm (Finsupp.single i (r₁ * r₂)) = Finsupp.single i.1 r₁ ⊗ₜ[R] Finsupp.single i.2 r₂
-- finsuppTensorFinsupp'_symm_single_eq_single_one_tmul
theorem finsuppTensorFinsupp'_symm_single_eq_single_one_tmul (i : ι × κ) (r : R) :
    (finsuppTensorFinsupp' R ι κ).symm (Finsupp.single i r) =
      Finsupp.single i.1 1 ⊗ₜ Finsupp.single i.2 r 
Arguments: (R : Type u_1) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] (i : ι × κ) (r : R)
Goal: (finsuppTensorFinsupp' R ι κ).symm (Finsupp.single i r) = Finsupp.single i.1 1 ⊗ₜ[R] Finsupp.single i.2 r
-- finsuppTensorFinsupp'_symm_single_eq_tmul_single_one
theorem finsuppTensorFinsupp'_symm_single_eq_tmul_single_one (i : ι × κ) (r : R) :
    (finsuppTensorFinsupp' R ι κ).symm (Finsupp.single i r) =
      Finsupp.single i.1 r ⊗ₜ Finsupp.single i.2 1 
Arguments: (R : Type u_1) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] (i : ι × κ) (r : R)
Goal: (finsuppTensorFinsupp' R ι κ).symm (Finsupp.single i r) = Finsupp.single i.1 r ⊗ₜ[R] Finsupp.single i.2 1
-- finsuppTensorFinsuppLid_self
theorem finsuppTensorFinsuppLid_self :
    finsuppTensorFinsuppLid R R ι κ = finsuppTensorFinsupp' R ι κ 
Arguments: (R : Type u_1) (ι : Type u_5) (κ : Type u_6) [CommSemiring R]
Goal: finsuppTensorFinsuppLid R R ι κ = finsuppTensorFinsupp' R ι κ
-- finsuppTensorFinsuppRid_self
theorem finsuppTensorFinsuppRid_self :
    finsuppTensorFinsuppRid R R ι κ = finsuppTensorFinsupp' R ι κ 
Arguments: (R : Type u_1) (ι : Type u_5) (κ : Type u_6) [CommSemiring R]
Goal: finsuppTensorFinsuppRid R R ι κ = finsuppTensorFinsupp' R ι κ
-- Projectivization.independent_iff
theorem independent_iff : Independent f ↔ LinearIndependent K (Projectivization.rep ∘ f) 
Arguments: {ι : Type u_1} {K : Type u_2} {V : Type u_3} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ℙ K V}
Goal: Projectivization.Independent f ↔ LinearIndependent K (Projectivization.rep ∘ f)
-- Projectivization.independent_iff_completeLattice_independent
theorem independent_iff_completeLattice_independent :
    Independent f ↔ CompleteLattice.Independent fun i => (f i).submodule 
Arguments: {ι : Type u_1} {K : Type u_2} {V : Type u_3} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ℙ K V}
Goal: Projectivization.Independent f ↔ CompleteLattice.Independent fun i => (f i).submodule
-- Projectivization.dependent_iff
theorem dependent_iff : Dependent f ↔ ¬LinearIndependent K (Projectivization.rep ∘ f) 
Arguments: {ι : Type u_1} {K : Type u_2} {V : Type u_3} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ℙ K V}
Goal: Projectivization.Dependent f ↔ ¬LinearIndependent K (Projectivization.rep ∘ f)
-- Projectivization.dependent_iff_not_independent
theorem dependent_iff_not_independent : Dependent f ↔ ¬Independent f 
Arguments: {ι : Type u_1} {K : Type u_2} {V : Type u_3} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ℙ K V}
Goal: Projectivization.Dependent f ↔ ¬Projectivization.Independent f
-- Projectivization.independent_iff_not_dependent
theorem independent_iff_not_dependent : Independent f ↔ ¬Dependent f 
Arguments: {ι : Type u_1} {K : Type u_2} {V : Type u_3} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ℙ K V}
Goal: Projectivization.Independent f ↔ ¬Projectivization.Dependent f
-- Projectivization.dependent_pair_iff_eq
@[simp]
theorem dependent_pair_iff_eq (u v : ℙ K V) : Dependent ![u, v] ↔ u = v 
Arguments: {K : Type u_2} {V : Type u_3} [DivisionRing K] [AddCommGroup V] [Module K V] (u v : ℙ K V)
Goal: Projectivization.Dependent ![u, v] ↔ u = v