MeasurableEquiv.prodUnique_symm_apply
No docstring.
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α : Type u_1MeasurableSpace αA measurable space is a space equipped with a σ-algebra. -
β : Type u_2MeasurableSpace βUnique βUnique αexpresses thatαis a type with a unique termdefault.
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a : α
(prodUnique α β).symm a = (a, default)MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
Unique : Sort u → Sort (max 1 u)`Unique α` expresses that `α` is a type with a unique term `default`. This is implemented as a type, rather than a `Prop`-valued predicate, for good definitional properties of the default term.
Eq : {α : Sort u_1} → α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
Conventions for notations in identifiers:
* The recommended spelling of `=` in identifiers is `eq`.MeasurableEquiv.prodUnique : (α : Type u_1) → (β : Type u_2) → [inst : MeasurableSpace α] → [inst_1 : MeasurableSpace β] → [Unique β] → α × β ≃ᵐ αThe measurable equivalence `α × β ≃ᵐ α` when `β` has a unique element.Go to its page
MeasurableEquiv.symm : {α : Type u_1} → {β : Type u_2} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSpace β] → α ≃ᵐ β → β ≃ᵐ αThe inverse of an equivalence between measurable spaces.
Prod.mk : {α : Type u} → {β : Type v} → α → β → α × βConstructs a pair. This is usually written `(x, y)` instead of `Prod.mk x y`. Conventions for notations in identifiers: * The recommended spelling of `(a, b)` in identifiers is `mk`.
Inhabited.default : {α : Sort u} → [self : Inhabited α] → α`default` is a function that produces a "default" element of any `Inhabited` type. This element does not have any particular specified properties, but it is often an all-zeroes value.
Code
lemma prodUnique_symm_apply {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] [Unique β]
(a : α) :
(prodUnique α β).symm a = (a, default)Proof
rfl
New in v4.34.0-rc2-39-gb743f31 (2026-09-08), and its meaning has not changed since.
Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.
Dependency graph
Audit surface: 1 project declarations, 16 external constants
✓ Proved: no sorry anywhere in its closure
This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.