Theorems · Theorem · functional analysis
Submodule.IsCompl.isTopCompl_iff_continuous_quotientEquivOfIsCompl
∀ {R : Type u_1} [inst : Ring R] {M : Type u_2} [inst_1 : TopologicalSpace M] [inst_2 : AddCommGroup M]
[inst_3 : Module R M] {p q : Submodule R M} [IsTopologicalAddGroup M] (h : IsCompl p q),
Submodule.IsTopCompl p q ↔ Continuous ⇑(p.quotientEquivOfIsCompl q h)Two complementary submodules are topological complements if and only if the linear equivalence
Submodule.quotientEquivOfIsCompl is continuous.
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- Foundations
- Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites20
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- DFunLike.coestatement and proof · cited by 62,936
- TopologicalSpacestatement and proof · cited by 24,529
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement · cited by 18,349
- AddCommGroupstatement and proof · cited by 12,871
- Ringstatement and proof · cited by 7,463
- Submodulestatement and proof · cited by 7,192
- LinearEquivstatement · cited by 3,317
- Continuousstatement and proof · cited by 2,592
- HasQuotient.Quotientstatement · cited by 2,301
- IsTopologicalAddGroupstatement and proof · cited by 1,394
- IsComplstatement and proof · cited by 351
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