MeasurableEquiv.coe_prodCongr
No docstring.
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α : Type u_2mα : MeasurableSpace αA measurable space is a space equipped with a σ-algebra. -
β : Type u_3mβ : MeasurableSpace β -
γ : Type u_4mγ : MeasurableSpace γ -
δ : Type u_5mδ : MeasurableSpace δ
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e₁ : α ≃ᵐ βEquivalences between measurable spaces. -
e₂ : γ ≃ᵐ δ
⇑(e₁.prodCongr e₂) = Prod.map ⇑e₁ ⇑e₂MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
MeasurableEquiv : (α : Type u_6) → (β : Type u_7) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_6 u_7)Equivalences between measurable spaces. Main application is the simplification of measurability statements along measurable equivalences.
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.prodCongr : {α : Type u_1} →
{β : Type u_2} →
{γ : Type u_3} →
{δ : Type u_4} →
[inst : MeasurableSpace α] →
[inst_1 : MeasurableSpace β] →
[inst_2 : MeasurableSpace γ] → [inst_3 : MeasurableSpace δ] → α ≃ᵐ β → γ ≃ᵐ δ → α × γ ≃ᵐ β × δProducts of equivalent measurable spaces are equivalent.
Prod.map : {α₁ : Type u₁} → {α₂ : Type u₂} → {β₁ : Type v₁} → {β₂ : Type v₂} → (α₁ → α₂) → (β₁ → β₂) → α₁ × β₁ → α₂ × β₂Transforms a pair by applying functions to both elements.
Examples:
* `(1, 2).map (· + 1) (· * 3) = (2, 6)`
* `(1, 2).map toString (· * 3) = ("1", 6)`Code
lemma coe_prodCongr {α β γ δ : Type*}
{mα : MeasurableSpace α} {mβ : MeasurableSpace β}
{mγ : MeasurableSpace γ} {mδ : MeasurableSpace δ}
(e₁ : MeasurableEquiv α β) (e₂ : MeasurableEquiv γ δ) :
(prodCongr e₁ e₂ : (α × γ) → (β × δ)) = Prod.map e₁ e₂Proof
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
Nothing to draw. Its statement rests on no other declaration in this project, and names nothing from a package left unaudited — so the graph is this declaration alone. That is the answer, not a missing picture.
Audit surface: 0 project declarations, 10 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.