[Merged by Bors] - refactor(topology/vector_bundle/basic): deduplicate trivialization
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…17359) Currently, in mathlib, there is a structure `topological_vector_bundle.trivialization` which extends another structure `topological_fibre_bundle.trivialization` by a linearity hypothesis. We change this to a single structure `trivialization` (no namespace), together with a mixin class `trivialization.is_linear`. This is the first step in the planned [vector bundle refactor](https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/Smooth.20vector.20bundles): it will allow all the data of a vector bundle to be held at the level of fibre bundles, so that there can be a non-"dangerous instance" `charted_space` instance associated to a vector bundle. The analogous change is made for `pretrivialization`. The most awkward aspect of this refactor is that one would want the linearity of trivializations associated to a vector bundle to be "automatic", which effectively means that one wants a `trivialization.is_linear` instance to be found by typeclass inference for trivializations in the `trivialization_atlas` of a vector bundle. For this to work, the property of belonging to the trivialization atlas also needs to be made into a Prop typeclass. We call this typeclass `mem_trivialization_atlas`. Co-authored-by: Floris van Doorn <fpvdoorn@gmail.com>
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…17359) Currently, in mathlib, there is a structure `topological_vector_bundle.trivialization` which extends another structure `topological_fibre_bundle.trivialization` by a linearity hypothesis. We change this to a single structure `trivialization` (no namespace), together with a mixin class `trivialization.is_linear`. This is the first step in the planned [vector bundle refactor](https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/Smooth.20vector.20bundles): it will allow all the data of a vector bundle to be held at the level of fibre bundles, so that there can be a non-"dangerous instance" `charted_space` instance associated to a vector bundle. The analogous change is made for `pretrivialization`. The most awkward aspect of this refactor is that one would want the linearity of trivializations associated to a vector bundle to be "automatic", which effectively means that one wants a `trivialization.is_linear` instance to be found by typeclass inference for trivializations in the `trivialization_atlas` of a vector bundle. For this to work, the property of belonging to the trivialization atlas also needs to be made into a Prop typeclass. We call this typeclass `mem_trivialization_atlas`. Co-authored-by: Floris van Doorn <fpvdoorn@gmail.com>
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) Split the file `topology.fiber_bundle` into two, `topology.fiber_bundle.trivialization` and `topology.fiber_bundle.basic`, the former treating (pre)trivializations and the latter treating fiber bundles and constructions for them (the core construction, the prebundle construction, etc). Also move lemmas from `topology.vector_bundle.basic` which turned out not to invove the linear structure (cf [this discussion](#17359 (comment))) into `topology.fiber_bundle.trivialization`.
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Currently, in mathlib, there is a structure
topological_vector_bundle.trivializationwhich extends another structuretopological_fibre_bundle.trivializationby a linearity hypothesis. We change this to a single structuretrivialization(no namespace), together with a mixin classtrivialization.is_linear. This is the first step in the planned vector bundle refactor: it will allow all the data of a vector bundle to be held at the level of fibre bundles, so that there can be a non-"dangerous instance"charted_spaceinstance associated to a vector bundle.The analogous change is made for
pretrivialization.The most awkward aspect of this refactor is that one would want the linearity of trivializations associated to a vector bundle to be "automatic", which effectively means that one wants a
trivialization.is_linearinstance to be found by typeclass inference for trivializations in thetrivialization_atlasof a vector bundle. For this to work, the property of belonging to the trivialization atlas also needs to be made into a Prop typeclass. We call this typeclassmem_trivialization_atlas.Co-authored-by: Floris van Doorn fpvdoorn@gmail.com