[Merged by Bors] - Feat (GroupTheory/GroupAction/Hom/Pointwise) : generalize smul set lemmas to group actions #12023
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Please do with my comments as you see fit. LGTM bors d+ |
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…mmas to group actions (#12023) This is a generalization of `Mathlib/Algebra/Module/LinearMap/Pointwise.lean` from `LinearMapClass` to `MulActionHomClass`. The preexisting lemmas are generalized. - `image_smul_setₛₗ` : under a `σ`-equivariant map, one has `h '' (c • s) = (σ c) • h '' s`. - `preimage_smul_setₛₗ'` is a general version of the equality `h ⁻¹' (σ c • s) = c • h⁻¹' s`. It requires that `c` acts surjectively and `σ c` acts injectively. It is provided with specific versions: - `preimage_smul_setₛₗ_of_units` requires that `c` and `σ c` are units - `MonoidHom.preimage_smul_setₛₗ` requires that `σ` is a `MonoidHom` and `c` is a unit - `MonoidHomClass.preimage_smul_setₛₗ` requires that `σ` belongs to a `MonoidHomClass`and that `c` is a unit - `Group.preimage_smul_setₛₗ` requires that the types of `c` and `σ c` are groups - `image_smul_set`, `preimage_smul_set` and `Group.preimage_smul_set` are the variants when `σ` is the identity.
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…mmas to group actions (#12023) This is a generalization of `Mathlib/Algebra/Module/LinearMap/Pointwise.lean` from `LinearMapClass` to `MulActionHomClass`. The preexisting lemmas are generalized. - `image_smul_setₛₗ` : under a `σ`-equivariant map, one has `h '' (c • s) = (σ c) • h '' s`. - `preimage_smul_setₛₗ'` is a general version of the equality `h ⁻¹' (σ c • s) = c • h⁻¹' s`. It requires that `c` acts surjectively and `σ c` acts injectively. It is provided with specific versions: - `preimage_smul_setₛₗ_of_units` requires that `c` and `σ c` are units - `MonoidHom.preimage_smul_setₛₗ` requires that `σ` is a `MonoidHom` and `c` is a unit - `MonoidHomClass.preimage_smul_setₛₗ` requires that `σ` belongs to a `MonoidHomClass`and that `c` is a unit - `Group.preimage_smul_setₛₗ` requires that the types of `c` and `σ c` are groups - `image_smul_set`, `preimage_smul_set` and `Group.preimage_smul_set` are the variants when `σ` is the identity.
atarnoam
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…mmas to group actions (#12023) This is a generalization of `Mathlib/Algebra/Module/LinearMap/Pointwise.lean` from `LinearMapClass` to `MulActionHomClass`. The preexisting lemmas are generalized. - `image_smul_setₛₗ` : under a `σ`-equivariant map, one has `h '' (c • s) = (σ c) • h '' s`. - `preimage_smul_setₛₗ'` is a general version of the equality `h ⁻¹' (σ c • s) = c • h⁻¹' s`. It requires that `c` acts surjectively and `σ c` acts injectively. It is provided with specific versions: - `preimage_smul_setₛₗ_of_units` requires that `c` and `σ c` are units - `MonoidHom.preimage_smul_setₛₗ` requires that `σ` is a `MonoidHom` and `c` is a unit - `MonoidHomClass.preimage_smul_setₛₗ` requires that `σ` belongs to a `MonoidHomClass`and that `c` is a unit - `Group.preimage_smul_setₛₗ` requires that the types of `c` and `σ c` are groups - `image_smul_set`, `preimage_smul_set` and `Group.preimage_smul_set` are the variants when `σ` is the identity.
uniwuni
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Apr 19, 2024
…mmas to group actions (#12023) This is a generalization of `Mathlib/Algebra/Module/LinearMap/Pointwise.lean` from `LinearMapClass` to `MulActionHomClass`. The preexisting lemmas are generalized. - `image_smul_setₛₗ` : under a `σ`-equivariant map, one has `h '' (c • s) = (σ c) • h '' s`. - `preimage_smul_setₛₗ'` is a general version of the equality `h ⁻¹' (σ c • s) = c • h⁻¹' s`. It requires that `c` acts surjectively and `σ c` acts injectively. It is provided with specific versions: - `preimage_smul_setₛₗ_of_units` requires that `c` and `σ c` are units - `MonoidHom.preimage_smul_setₛₗ` requires that `σ` is a `MonoidHom` and `c` is a unit - `MonoidHomClass.preimage_smul_setₛₗ` requires that `σ` belongs to a `MonoidHomClass`and that `c` is a unit - `Group.preimage_smul_setₛₗ` requires that the types of `c` and `σ c` are groups - `image_smul_set`, `preimage_smul_set` and `Group.preimage_smul_set` are the variants when `σ` is the identity.
callesonne
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Apr 22, 2024
…mmas to group actions (#12023) This is a generalization of `Mathlib/Algebra/Module/LinearMap/Pointwise.lean` from `LinearMapClass` to `MulActionHomClass`. The preexisting lemmas are generalized. - `image_smul_setₛₗ` : under a `σ`-equivariant map, one has `h '' (c • s) = (σ c) • h '' s`. - `preimage_smul_setₛₗ'` is a general version of the equality `h ⁻¹' (σ c • s) = c • h⁻¹' s`. It requires that `c` acts surjectively and `σ c` acts injectively. It is provided with specific versions: - `preimage_smul_setₛₗ_of_units` requires that `c` and `σ c` are units - `MonoidHom.preimage_smul_setₛₗ` requires that `σ` is a `MonoidHom` and `c` is a unit - `MonoidHomClass.preimage_smul_setₛₗ` requires that `σ` belongs to a `MonoidHomClass`and that `c` is a unit - `Group.preimage_smul_setₛₗ` requires that the types of `c` and `σ c` are groups - `image_smul_set`, `preimage_smul_set` and `Group.preimage_smul_set` are the variants when `σ` is the identity.
Jun2M
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Apr 24, 2024
…mmas to group actions (#12023) This is a generalization of `Mathlib/Algebra/Module/LinearMap/Pointwise.lean` from `LinearMapClass` to `MulActionHomClass`. The preexisting lemmas are generalized. - `image_smul_setₛₗ` : under a `σ`-equivariant map, one has `h '' (c • s) = (σ c) • h '' s`. - `preimage_smul_setₛₗ'` is a general version of the equality `h ⁻¹' (σ c • s) = c • h⁻¹' s`. It requires that `c` acts surjectively and `σ c` acts injectively. It is provided with specific versions: - `preimage_smul_setₛₗ_of_units` requires that `c` and `σ c` are units - `MonoidHom.preimage_smul_setₛₗ` requires that `σ` is a `MonoidHom` and `c` is a unit - `MonoidHomClass.preimage_smul_setₛₗ` requires that `σ` belongs to a `MonoidHomClass`and that `c` is a unit - `Group.preimage_smul_setₛₗ` requires that the types of `c` and `σ c` are groups - `image_smul_set`, `preimage_smul_set` and `Group.preimage_smul_set` are the variants when `σ` is the identity.
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This is a generalization of
Mathlib/Algebra/Module/LinearMap/Pointwise.leanfromLinearMapClasstoMulActionHomClass.The preexisting lemmas are generalized.
image_smul_setₛₗ: under aσ-equivariant map,one has
h '' (c • s) = (σ c) • h '' s.preimage_smul_setₛₗ'is a general version of the equalityh ⁻¹' (σ c • s) = c • h⁻¹' s. It requires thatcacts surjectively andσ cacts injectively.It is provided with specific versions:
preimage_smul_setₛₗ_of_unitsrequires thatcandσ care unitsMonoidHom.preimage_smul_setₛₗrequires thatσis aMonoidHomandcis a unitMonoidHomClass.preimage_smul_setₛₗrequires thatσbelongs to aMonoidHomClassand thatcis a unitGroup.preimage_smul_setₛₗrequires that the types ofcandσ care groupsimage_smul_set,preimage_smul_setandGroup.preimage_smul_setarethe variants when
σis the identity.