Theorems · Theorem · linear algebra
Submodule.rank_quotient_add_rank
∀ {R : Type u_1} {M : Type u} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
[HasRankNullity.{u, u_1} R] (N : Submodule R M), Module.rank R (M ⧸ N) + Module.rank R ↥N = Module.rank R M- Cited by
- 8 results in Mathlib
- Foundations
- Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- Ringstatement and proof · cited by 7,463
- Submodulestatement and proof · cited by 7,192
- Cardinalstatement · cited by 2,598
- HasQuotient.Quotientstatement · cited by 2,301
- Module.rankstatement · cited by 496
- HasRankNullitystatement and proof · cited by 26
- HasRankNullity.rank_quotient_add_rankproof · cited by 2
Cited by8
Results whose statement or proof uses this declaration.
- ModularForm.rank_eq_one_add_rank_cuspFormproof · cited by 3
- Submodule.rank_sup_add_rank_inf_eqproof · cited by 2
- LinearMap.lift_rank_range_add_rank_kerproof · cited by 2
- exists_linearIndepOn_of_lt_rankproof · cited by 1
- Ideal.rank_pow_quot_auxproof · cited by 1
- LinearMap.rank_range_add_rank_kerproof · cited by 1
- nontrivial_of_hasRankNullityproof · cited by 0
- Submodule.exists_smul_notMem_of_rank_ltproof · cited by 0