File: Mathlib/Algebra/MonoidAlgebra/Support.lean

full_name: MonoidAlgebra.support_mul
code: theorem support_mul [Mul G] [DecidableEq G] (a b : MonoidAlgebra k G) :
    (a * b).support ⊆ a.support * b.support 
goal: (a * b).support ⊆ a.support * b.support

full_name: MonoidAlgebra.support_single_mul_subset
code: theorem support_single_mul_subset [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) (r : k) (a : G) :
    (single a r * f : MonoidAlgebra k G).support ⊆ Finset.image (a * ·) f.support 
goal: (MonoidAlgebra.single a r * f).support ⊆ Finset.image (fun x => a * x) f.support

full_name: MonoidAlgebra.support_mul_single_subset
code: theorem support_mul_single_subset [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) (r : k) (a : G) :
    (f * single a r).support ⊆ Finset.image (· * a) f.support 
goal: (f * MonoidAlgebra.single a r).support ⊆ Finset.image (fun x => x * a) f.support

full_name: MonoidAlgebra.support_single_mul_eq_image
code: theorem support_single_mul_eq_image [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) {r : k}
    (hr : ∀ y, r * y = 0 ↔ y = 0) {x : G} (lx : IsLeftRegular x) :
    (single x r * f : MonoidAlgebra k G).support = Finset.image (x * ·) f.support 
goal: (MonoidAlgebra.single x r * f).support = Finset.image (fun x_1 => x * x_1) f.support

full_name: MonoidAlgebra.support_mul_single_eq_image
code: theorem support_mul_single_eq_image [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) {r : k}
    (hr : ∀ y, y * r = 0 ↔ y = 0) {x : G} (rx : IsRightRegular x) :
    (f * single x r).support = Finset.image (· * x) f.support 
goal: (f * MonoidAlgebra.single x r).support = Finset.image (fun x_1 => x_1 * x) f.support

full_name: MonoidAlgebra.support_mul_single
code: theorem support_mul_single [Mul G] [IsRightCancelMul G] (f : MonoidAlgebra k G) (r : k)
    (hr : ∀ y, y * r = 0 ↔ y = 0) (x : G) :
    (f * single x r).support = f.support.map (mulRightEmbedding x) 
goal: (f * MonoidAlgebra.single x r).support = Finset.map (mulRightEmbedding x) f.support

full_name: MonoidAlgebra.support_single_mul
code: theorem support_single_mul [Mul G] [IsLeftCancelMul G] (f : MonoidAlgebra k G) (r : k)
    (hr : ∀ y, r * y = 0 ↔ y = 0) (x : G) :
    (single x r * f : MonoidAlgebra k G).support = f.support.map (mulLeftEmbedding x) 
goal: (MonoidAlgebra.single x r * f).support = Finset.map (mulLeftEmbedding x) f.support

full_name: MonoidAlgebra.support_one_subset
code: lemma support_one_subset [One G] : (1 : MonoidAlgebra k G).support ⊆ 1 
goal: Finsupp.support 1 ⊆ 1

full_name: MonoidAlgebra.support_one
code: @[simp]
lemma support_one [One G] [NeZero (1 : k)] : (1 : MonoidAlgebra k G).support = 1 
goal: Finsupp.support 1 = 1

full_name: MonoidAlgebra.mem_span_support
code: /-- An element of `MonoidAlgebra k G` is in the subalgebra generated by its support. -/
theorem mem_span_support (f : MonoidAlgebra k G) :
    f ∈ Submodule.span k (of k G '' (f.support : Set G)) 
goal: f ∈ Submodule.span k (⇑(MonoidAlgebra.of k G) '' ↑f.support)

full_name: AddMonoidAlgebra.support_mul
code: theorem support_mul [DecidableEq G] [Add G] (a b : k[G]) :
    (a * b).support ⊆ a.support + b.support 
goal: (a * b).support ⊆ a.support + b.support

full_name: AddMonoidAlgebra.support_mul_single
code: theorem support_mul_single [Add G] [IsRightCancelAdd G] (f : k[G]) (r : k)
    (hr : ∀ y, y * r = 0 ↔ y = 0) (x : G) :
    (f * single x r : k[G]).support = f.support.map (addRightEmbedding x) 
goal: (f * AddMonoidAlgebra.single x r).support = Finset.map (addRightEmbedding x) f.support

full_name: AddMonoidAlgebra.support_single_mul
code: theorem support_single_mul [Add G] [IsLeftCancelAdd G] (f : k[G]) (r : k)
    (hr : ∀ y, r * y = 0 ↔ y = 0) (x : G) :
    (single x r * f : k[G]).support = f.support.map (addLeftEmbedding x) 
goal: (AddMonoidAlgebra.single x r * f).support = Finset.map (addLeftEmbedding x) f.support

full_name: AddMonoidAlgebra.support_one_subset
code: lemma support_one_subset [Zero G] : (1 : k[G]).support ⊆ 0 
goal: Finsupp.support 1 ⊆ 0

full_name: AddMonoidAlgebra.support_one
code: @[simp]
lemma support_one [Zero G] [NeZero (1 : k)] : (1 : k[G]).support = 0 
goal: Finsupp.support 1 = 0

full_name: AddMonoidAlgebra.mem_span_support
code: /-- An element of `k[G]` is in the submodule generated by its support. -/
theorem mem_span_support [AddZeroClass G] (f : k[G]) :
    f ∈ Submodule.span k (of k G '' (f.support : Set G)) 
goal: f ∈ Submodule.span k (⇑(AddMonoidAlgebra.of k G) '' ↑f.support)

full_name: AddMonoidAlgebra.mem_span_support'
code: /-- An element of `k[G]` is in the subalgebra generated by its support, using
unbundled inclusion. -/
theorem mem_span_support' (f : k[G]) :
    f ∈ Submodule.span k (of' k G '' (f.support : Set G)) 
goal: f ∈ Submodule.span k (AddMonoidAlgebra.of' k G '' ↑f.support)