Theorems · Theorem · ring theory
OrthogonalIdempotents.prod_one_sub
∀ {R : Type u_1} [inst : CommRing R] {I : Type u_3} {e : I → R},
OrthogonalIdempotents e → ∀ (s : Finset I), ∏ i ∈ s, (1 - e i) = 1 - ∑ i ∈ s, e i- Defined in
- Mathlib.RingTheory.Idempotents
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Finsetstatement and proof · cited by 13,712
- Finset.sumstatement and proof · cited by 5,195
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
- Finset.prodstatement and proof · cited by 2,356
- sub_zeroproof · cited by 938
- mul_subproof · cited by 201
- sub_mulproof · cited by 170
- Finset.cons_inductionproof · cited by 85
- Finset.sum_consproof · cited by 84
- Finset.prod_consproof · cited by 60
Cited by2
Results whose statement or proof uses this declaration.
- CompleteOrthogonalIdempotents.prod_one_subproof · cited by 1
- CompleteOrthogonalIdempotents.of_prod_one_subproof · cited by 1