ContinuousSMulDiscrete.lean`

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6 declarations, 0 contain sorry, 0 declared as axioms

Continuous actions on discrete modules

The typeclass ContinuousSMulDiscrete G M, expressing that a G-action on a (potentially untopologized) M is discretely continuous: every stabiliser is open. Equivalent to ContinuousSMul once M is given the discrete topology.

Material destined for Mathlib.

Imports (2)

Imported by 4 files

class ContinuousSMulDiscrete ContinuousSMulDiscrete G M means that the action G × M → M is continuous when M is given the discrete topology (See continuousSMulDiscrete_iff) line 32 source
Source documentation
ContinuousSMulDiscrete G M means that the action G × M → M is continuous when M is given the discrete topology (See continuousSMulDiscrete_iff).
This class is especially useful when M already has another topology and we cannot easily put the discrete topology on it.

no axioms what this means
Statement uses: TopologicalSpace, SMul
```
class ContinuousSMulDiscrete (G M : Type*) [TopologicalSpace G] [SMul G M] : Prop where
…
```
lemma continuousSMulDiscrete_iff line 35 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, SMul, DiscreteTopology, Iff, ContinuousSMulDiscrete, ContinuousSMul
Proof uses: Iff.intro, ContinuousSMul.mk, Iff.mpr, Continuous, Prod, HSMul.hSMul, Prod.fst, Prod.snd, IsOpen, Set.preimage, Singleton.singleton, Set, continuous_discrete_rng, Eq.mpr, Set.iUnion, SProd.sprod, Set.ofPred, Eq, eq_of_heq, HEq and 24 more
12 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
instance (instance at line 44) line 44 source
lemma ContinuousSMulDiscrete.of_continuousSMul line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, SMul, DiscreteTopology, ContinuousSMul, ContinuousSMulDiscrete
Proof uses: Eq.mpr, id, Eq, congrArg, propext, continuousSMulDiscrete_iff

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma continuousSMulDiscrete_iff_isOpen_stabilizer line 52 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, Group, ContinuousMul, MulAction, Iff, ContinuousSMulDiscrete, IsOpen, Subgroup, MulAction.stabilizer
Proof uses: Iff.intro, ContinuousSMulDiscrete.isOpen_smul_eq, ContinuousSMulDiscrete.mk, Eq, Set, Set.ofPred, HSMul.hSMul, EmptyCollection.emptyCollection, Set.Nonempty, Or, Set.eq_empty_or_nonempty, Eq.rec, isOpen_empty, Eq.symm, Membership.mem, Eq.mpr, Set.preimage, Homeomorph, Homeomorph.mulLeft, Inv.inv and 15 more
41 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
lemma ContinuousSMulDiscrete.isOpen_stabilizer line 62 source
standard axioms: propext what this means
Statement uses: TopologicalSpace, Group, MulAction, ContinuousSMulDiscrete, IsOpen, Subgroup, MulAction.stabilizer
Proof uses: ContinuousSMulDiscrete.isOpen_smul_eq
12 instances, coercions and auxiliary constants left out.