Theorems · Theorem · ring theory
eq_isotypicComponent_iff
∀ {R : Type u_2} {M : Type u} {S : Type u_4} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : AddCommGroup S]
[inst_3 : Module R M] [inst_4 : Module R S] [IsSimpleModule R S] [IsSemisimpleModule R M] {m : Submodule R M},
m ≠ ⊥ → (m = isotypicComponent R M S ↔ IsIsotypicOfType R (↥m) S ∧ m.IsFullyInvariant)- Defined in
- Mathlib.RingTheory.SimpleModule.Isotypic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
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
- Bot.botstatement and proof · cited by 4,720
- LinearEquiv.symmproof · cited by 1,461
- LE.le.antisymmproof · cited by 507
- Nonempty.someproof · cited by 340
- Eq.trans_leproof · cited by 155
- IsSimpleModulestatement and proof · cited by 114
- IsSemisimpleModulestatement and proof · cited by 67
- isotypicComponentstatement and proof · cited by 21
Cited by1
Results whose statement or proof uses this declaration.
- mem_isotypicComponents_iffproof · cited by 0