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import Mathlib.Algebra.Group.Pointwise.Set | ||
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open scoped Pointwise | ||
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namespace Set | ||
variable {α β : Type*} [SMul α β] | ||
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attribute [gcongr] smul_subset_smul vadd_subset_vadd smul_set_mono vadd_set_mono | ||
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@[to_additive] | ||
lemma smul_set_insert (a : α) (b : β) (s : Set β) : a • insert b s = insert (a • b) (a • s) := | ||
image_insert_eq .. | ||
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end Set |
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LeanCamCombi/Mathlib/Algebra/GroupWithZero/Pointwise/Set.lean
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import Mathlib.Algebra.Group.Pointwise.Finset.NatCard | ||
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open scoped Cardinal Pointwise | ||
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variable {G α G₀ M₀ : Type*} | ||
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namespace Set | ||
section Group | ||
variable [Group G] [MulAction G α] | ||
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@[to_additive (attr := simp)] | ||
lemma card_smul_set' (a : G) (s : Set α) : #↥(a • s) = #s := | ||
Cardinal.mk_image_eq_of_injOn _ _ (MulAction.injective a).injOn | ||
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end Group | ||
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section GroupWithZero | ||
variable [GroupWithZero G₀] [Zero M₀] [MulActionWithZero G₀ M₀] {a : G₀} | ||
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lemma card_smul_set₀ (ha : a ≠ 0) (s : Set M₀) : Nat.card ↥(a • s) = Nat.card s := | ||
Nat.card_image_of_injective (MulAction.injective₀ ha) _ | ||
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end GroupWithZero |
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import Mathlib.Data.Set.Card | ||
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/-! | ||
# TODO | ||
Rename `Set.exists_subset_card_eq` to `Set.exists_subset_ncard_eq` | ||
-/ | ||
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open scoped Cardinal | ||
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namespace Set | ||
variable {α : Type*} {s : Set α} | ||
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lemma exists_subset_natCard_eq {c : Cardinal} (hc : c ≤ #s) : ∃ t ⊆ s, #t = c := | ||
Cardinal.le_mk_iff_exists_subset.mp hc | ||
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end Set |
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40
LeanCamCombi/Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
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import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace | ||
import Mathlib.LinearAlgebra.AffineSpace.Combination | ||
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | ||
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/-! | ||
# TODO | ||
Kill `spanPoints` | ||
-/ | ||
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open Set | ||
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variable {k V P : Type*} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] | ||
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namespace AffineSubspace | ||
variable {s : AffineSubspace k P} {x y z : P} | ||
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lemma mem_of_collinear (h : Collinear k {x, y, z}) (hx : x ∈ s) (hz : z ∈ s) : y ∈ s := sorry | ||
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end AffineSubspace | ||
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-- TODO: Prove that `SameRay` implies `Collinear` | ||
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@[simp] lemma affineSpan_insert_zero (s : Set V) : | ||
(affineSpan k (insert 0 s) : Set V) = Submodule.span k s := by | ||
refine subset_antisymm ?_ ?_ | ||
· rw [← Submodule.span_insert_zero] | ||
exact affineSpan_subset_span | ||
let W : Submodule k V := | ||
{ carrier := affineSpan k (insert 0 s) | ||
add_mem' := fun {x y} hx hy ↦ by | ||
sorry | ||
zero_mem' := subset_affineSpan _ _ <| mem_insert .. | ||
smul_mem' := fun {a x} hx ↦ by | ||
simp only [SetLike.mem_coe] | ||
refine AffineSubspace.mem_of_collinear ?_ hx <| subset_affineSpan _ _ <| mem_insert .. | ||
sorry | ||
} | ||
change Submodule.span k s ≤ W | ||
exact Submodule.span_le.2 fun x hx ↦ subset_affineSpan _ _ <| subset_insert _ _ hx |
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LeanCamCombi/Mathlib/LinearAlgebra/Dimension/Construction.lean
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import Mathlib.LinearAlgebra.Dimension.Constructions | ||
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namespace Submodule | ||
variable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] {s t : Submodule R M} | ||
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-- TODO: Generalise `finrank_mono` | ||
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lemma rank_mono (hst : s ≤ t) : Module.rank R s ≤ Module.rank R t := rank_le_of_submodule _ _ hst | ||
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end Submodule |
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import Mathlib.LinearAlgebra.Span | ||
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attribute [gcongr] Submodule.span_mono |
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