Theorems · Theorem · ring theory
DirectSum.coe_mul_apply_eq_sum_antidiagonal
∀ {ι : Type u_1} {σ : Type u_2} {R : Type u_4} [inst : DecidableEq ι] [inst_1 : Semiring R] [inst_2 : SetLike σ R]
[inst_3 : AddSubmonoidClass σ R] (A : ι → σ) [inst_4 : AddMonoid ι] [inst_5 : Finset.HasAntidiagonal ι]
[inst_6 : SetLike.GradedMonoid A] (r r' : DirectSum ι fun i => ↥(A i)) (n : ι),
↑((r * r') n) = ∑ ij ∈ Finset.HasAntidiagonal.antidiagonal n, ↑(r ij.1) * ↑(r' ij.2)- Defined in
- Mathlib.Algebra.DirectSum.Internal
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
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Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- Finset.sumstatement and proof · cited by 5,195
- AddMonoidstatement and proof · cited by 2,864
- MulZeroClass.mul_zeroproof · cited by 2,091
- SProd.sprodproof · cited by 1,750
- MulZeroClass.zero_mulproof · cited by 1,625
- SetLikestatement and proof · cited by 1,084
- Finset.filterproof · cited by 949
- DFinsuppstatement · cited by 694
- DirectSumstatement and proof · cited by 446
- AddSubmonoidClassstatement and proof · cited by 346
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