Theorems · Theorem · general topology
Continuous.fst
∀ {X : Type u} {Y : Type v} {Z : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y]
[inst_2 : TopologicalSpace Z] {f : X → Y × Z}, Continuous f → Continuous fun x => (f x).1Postcomposing f with Prod.fst is continuous
- Defined in
- Mathlib.Topology.Constructions.SumProd
- Cited by
- 73 results in Mathlib
- Foundations
- Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopologicalSpacestatement and proof · cited by 24,529
- Continuousstatement and proof · cited by 2,592
- Continuous.compproof · cited by 371
- continuous_fstproof · cited by 103
Cited by73
Results whose statement or proof uses this declaration.
- isUniformAddGroup_of_addCommGroupproof · cited by 21
- IsScalarTower.continuousSMulproof · cited by 8
- Units.continuous_valproof · cited by 8
- IsBoundedBilinearMap.hasStrictFDerivAtproof · cited by 7
- IsClosed.vadd_left_of_isCompactproof · cited by 5
- AffineMap.lineMap_continuous_uncurryproof · cited by 5
- IsClosed.smul_left_of_isCompactproof · cited by 5
- ContinuousLinearMap.continuous_uncurry_of_multilinearproof · cited by 4
- isUniformGroup_of_commGroupproof · cited by 4
- Isometry.completion_extensionproof · cited by 4
- Measure.ext_of_integral_prod_mul_prod_boundedContinuousFunctionproof · cited by 4
- NumberField.mixedEmbedding.continuous_normAtPlaceproof · cited by 3