[Merged by Bors] - refactor(topology/uniform_space/basic): review API #18516
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### API about uniform embeddings * Add `mk_iff` to `uniform_inducing` and `uniform_embedding`. * Move lemmas about `uniform_inducing` up. * Add `uniform_inducing.comap_uniform_space`, `uniform_inducing_iff'`, and `filter.has_basis.uniform_inducing_iff`. * Add `uniform_embedding_iff'`, `filter.has_basis.uniform_embedding_iff'`, and `filter.has_basis.uniform_embedding_iff`. * Drop `uniform_embedding_def` and `uniform_embedding_def'`. * Add `uniform_embedding_iff_uniform_inducing`. ### Other changes * Add `rescale_to_shell_semi_normed_zpow` and `rescale_to_shell_zpow`. * Generalize `continuous_linear_map.antilipschitz_of_uniform_embedding` to `continuous_linear_map.antilipschitz_of_embedding`, add an even more general version `linear_map.antilipschitz_of_comap_nhds_le`. * Use `fully_applied := ff` to generate `equiv.prod_congr_apply`. * Use `edist := λ _ _, 0` in `metric_space` instances for `empty` and `punit`. * Add `inducing.injective`, `inducing.embedding`, and `embedding_iff_inducing` * Allow `Sort*`s in `filter.has_basis.uniform_continuous_iff` and `filter.has_basis.uniform_continuous_on_iff`. * Rename * `metric.of_t0_pseudo_metric_space` to `metric_space.of_t0_pseudo_metric_space`; * `emetric.of_t0_pseudo_emetric_space` to `emetric_space.of_t0_pseudo_emetric_space`; * `metric.metric_space.to_emetric_space` to `metric_space.to_emetric_space`; * `uniform_embedding_iff'` to `emetric.uniform_embedding_iff'`
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Forward-port leanprover-community/mathlib#18516 * [`logic.equiv.basic`@`d6aae1bcbd04b8de2022b9b83a5b5b10e10c777d`..`195fcd60ff2bfe392543bceb0ec2adcdb472db4c`](https://leanprover-community.github.io/mathlib-port-status/file/logic/equiv/basic?range=d6aae1bcbd04b8de2022b9b83a5b5b10e10c777d..195fcd60ff2bfe392543bceb0ec2adcdb472db4c) * [`topology.separation`@`92ca63f0fb391a9ca5f22d2409a6080e786d99f7`..`195fcd60ff2bfe392543bceb0ec2adcdb472db4c`](https://leanprover-community.github.io/mathlib-port-status/file/topology/separation?range=92ca63f0fb391a9ca5f22d2409a6080e786d99f7..195fcd60ff2bfe392543bceb0ec2adcdb472db4c) * [`topology.uniform_space.basic`@`e1a7bdeb4fd826b7e71d130d34988f0a2d26a177`..`195fcd60ff2bfe392543bceb0ec2adcdb472db4c`](https://leanprover-community.github.io/mathlib-port-status/file/topology/uniform_space/basic?range=e1a7bdeb4fd826b7e71d130d34988f0a2d26a177..195fcd60ff2bfe392543bceb0ec2adcdb472db4c) * [`topology.uniform_space.uniform_embedding`@`2705404e701abc6b3127da906f40bae062a169c9`..`195fcd60ff2bfe392543bceb0ec2adcdb472db4c`](https://leanprover-community.github.io/mathlib-port-status/file/topology/uniform_space/uniform_embedding?range=2705404e701abc6b3127da906f40bae062a169c9..195fcd60ff2bfe392543bceb0ec2adcdb472db4c)
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Forward-port leanprover-community/mathlib#18516 * [`logic.equiv.basic`@`d6aae1bcbd04b8de2022b9b83a5b5b10e10c777d`..`195fcd60ff2bfe392543bceb0ec2adcdb472db4c`](https://leanprover-community.github.io/mathlib-port-status/file/logic/equiv/basic?range=d6aae1bcbd04b8de2022b9b83a5b5b10e10c777d..195fcd60ff2bfe392543bceb0ec2adcdb472db4c) * [`topology.separation`@`92ca63f0fb391a9ca5f22d2409a6080e786d99f7`..`195fcd60ff2bfe392543bceb0ec2adcdb472db4c`](https://leanprover-community.github.io/mathlib-port-status/file/topology/separation?range=92ca63f0fb391a9ca5f22d2409a6080e786d99f7..195fcd60ff2bfe392543bceb0ec2adcdb472db4c) * [`topology.uniform_space.basic`@`e1a7bdeb4fd826b7e71d130d34988f0a2d26a177`..`195fcd60ff2bfe392543bceb0ec2adcdb472db4c`](https://leanprover-community.github.io/mathlib-port-status/file/topology/uniform_space/basic?range=e1a7bdeb4fd826b7e71d130d34988f0a2d26a177..195fcd60ff2bfe392543bceb0ec2adcdb472db4c) * [`topology.uniform_space.uniform_embedding`@`2705404e701abc6b3127da906f40bae062a169c9`..`195fcd60ff2bfe392543bceb0ec2adcdb472db4c`](https://leanprover-community.github.io/mathlib-port-status/file/topology/uniform_space/uniform_embedding?range=2705404e701abc6b3127da906f40bae062a169c9..195fcd60ff2bfe392543bceb0ec2adcdb472db4c)
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API about uniform embeddings
mk_ifftouniform_inducinganduniform_embedding.uniform_inducingup.uniform_inducing.comap_uniform_space,uniform_inducing_iff', andfilter.has_basis.uniform_inducing_iff.uniform_embedding_iff',filter.has_basis.uniform_embedding_iff', andfilter.has_basis.uniform_embedding_iff.uniform_embedding_defanduniform_embedding_def'.uniform_embedding_iff_uniform_inducing.Other changes
rescale_to_shell_semi_normed_zpowandrescale_to_shell_zpow.continuous_linear_map.antilipschitz_of_uniform_embeddingtocontinuous_linear_map.antilipschitz_of_embedding, add an even more general versionlinear_map.antilipschitz_of_comap_nhds_le.fully_applied := ffto generateequiv.prod_congr_apply.edist := λ _ _, 0inmetric_spaceinstances foremptyandpunit.inducing.injective,inducing.embedding, andembedding_iff_inducingSort*s infilter.has_basis.uniform_continuous_iffandfilter.has_basis.uniform_continuous_on_iff.metric.of_t0_pseudo_metric_spacetometric_space.of_t0_pseudo_metric_space;emetric.of_t0_pseudo_emetric_spacetoemetric_space.of_t0_pseudo_emetric_space;metric.metric_space.to_emetric_spacetometric_space.to_emetric_space;uniform_embedding_iff'toemetric.uniform_embedding_iff'