Theorems · Theorem · global analysis
isOpenMap_of_hasStrictFDerivAt_equiv
∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] [CompleteSpace E]
{f : E → F} {f' : E → E ≃L[𝕜] F}, (∀ (x : E), HasStrictFDerivAt f (↑(f' x)) x) → IsOpenMap fIf a function has an invertible strict derivative at all points, then it is an open map.
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- Foundations
- Depth 186 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- RingHom.idstatement and proof · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- CompleteSpacestatement and proof · cited by 2,532
- ContinuousLinearEquivstatement and proof · cited by 743
- ContinuousLinearEquiv.toContinuousLinearMapstatement and proof · cited by 448
- Eq.geproof · cited by 375
- HasStrictFDerivAtstatement and proof · cited by 261
- IsOpenMapstatement · cited by 253
- isOpenMap_iff_nhds_leproof · cited by 6
- HasStrictFDerivAt.map_nhds_eq_of_equivproof · cited by 5
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