[Merged by Bors] - feat: TwoUniqueProds #7169
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Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>
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+ Introduce `TwoUniqueProds/Sums`: let `R[G]` be a monoid algebra over a semiring `R` without zero-divisors. A natural sufficient condition for `R[G]` to have no zero-divisors is that `G` has `UniqueProds`, as is shown by [MonoidAlgebra.instNoZeroDivisorsOfUniqueProds](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/NoZeroDivisors.html#MonoidAlgebra.instNoZeroDivisorsOfUniqueProds). Similarly, a natural sufficient condition for `R[G]` to have only trivial units (of the form `rg` with `r` a unit in `R` and `g` a unit in `G`) is that `G` has `TwoUniqueProds`, but we don't prove this yet in this PR. `TwoUniqueProds G` is also a natural sufficient condition in order for factors of a homogeneous element in an algebra graded by `G` without zero-divisors to themselves be homogeneous. + Show `TwoUniqueProds` implies `UniqueProds`: `TwoUniqueProds.toUniqueProds` + Strengthen `of_Covariant_right/left` to have `TwoUniqueProds` as conclusion + Extract `of_image_filter` from the proof of the instance `UniqueProds (∀ i, G i)` and use it also in the proof of `TwoUniqueProds (∀ i, G i)` + Use some private defs (starting from `private abbrev I`) to transfer `(Two)UniqueProds (∀ i, G i)` instances to `(Two)UniqueProds (G × H)` + Move the `[Module ℚ G] : UniqueSums G` instance from NoZeroDivisors.lean to UniqueProds.lean and strengthen to `TwoUniqueSums` New lemmas about UniqueMul: + `of_card_le_one`, `iff_card_le_one`, `UniqueMul_of_TwoUniqueMul` Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
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+ Introduce `TwoUniqueProds/Sums`: let `R[G]` be a monoid algebra over a semiring `R` without zero-divisors. A natural sufficient condition for `R[G]` to have no zero-divisors is that `G` has `UniqueProds`, as is shown by [MonoidAlgebra.instNoZeroDivisorsOfUniqueProds](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/NoZeroDivisors.html#MonoidAlgebra.instNoZeroDivisorsOfUniqueProds). Similarly, a natural sufficient condition for `R[G]` to have only trivial units (of the form `rg` with `r` a unit in `R` and `g` a unit in `G`) is that `G` has `TwoUniqueProds`, but we don't prove this yet in this PR. `TwoUniqueProds G` is also a natural sufficient condition in order for factors of a homogeneous element in an algebra graded by `G` without zero-divisors to themselves be homogeneous. + Show `TwoUniqueProds` implies `UniqueProds`: `TwoUniqueProds.toUniqueProds` + Strengthen `of_Covariant_right/left` to have `TwoUniqueProds` as conclusion + Extract `of_image_filter` from the proof of the instance `UniqueProds (∀ i, G i)` and use it also in the proof of `TwoUniqueProds (∀ i, G i)` + Use some private defs (starting from `private abbrev I`) to transfer `(Two)UniqueProds (∀ i, G i)` instances to `(Two)UniqueProds (G × H)` + Move the `[Module ℚ G] : UniqueSums G` instance from NoZeroDivisors.lean to UniqueProds.lean and strengthen to `TwoUniqueSums` New lemmas about UniqueMul: + `of_card_le_one`, `iff_card_le_one`, `UniqueMul_of_TwoUniqueMul` Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
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Introduce
TwoUniqueProds/Sums: letR[G]be a monoid algebra over a semiringRwithout zero-divisors. A natural sufficient condition forR[G]to have no zero-divisors is thatGhasUniqueProds, as is shown by MonoidAlgebra.instNoZeroDivisorsOfUniqueProds. Similarly, a natural sufficient condition forR[G]to have only trivial units (of the formrgwithra unit inRandga unit inG) is thatGhasTwoUniqueProds, but we don't prove this yet in this PR.TwoUniqueProds Gis also a natural sufficient condition in order for factors of a homogeneous element in an algebra graded byGwithout zero-divisors to themselves be homogeneous.Show
TwoUniqueProdsimpliesUniqueProds:TwoUniqueProds.toUniqueProdsStrengthen
of_Covariant_right/leftto haveTwoUniqueProdsas conclusionExtract
of_image_filterfrom the proof of the instanceUniqueProds (∀ i, G i)and use it also in the proof ofTwoUniqueProds (∀ i, G i)Use some private defs (starting from
private abbrev I) to transfer(Two)UniqueProds (∀ i, G i)instances to(Two)UniqueProds (G × H)Move the
[Module ℚ G] : UniqueSums Ginstance from NoZeroDivisors.lean to UniqueProds.lean and strengthen toTwoUniqueSumsNew lemmas about UniqueMul:
of_card_le_one,iff_card_le_one,UniqueMul_of_TwoUniqueMul