File: Mathlib/MeasureTheory/Group/MeasurableEquiv.lean

full_name: MeasurableEquiv.smul_toEquiv
code: /-- If a group `G` acts on `α` by measurable maps, then each element `c : G` defines a measurable
automorphism of `α`. -/
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
      "If an additive group `G` acts on `α` by measurable maps, then each element `c : G`
      defines a measurable automorphism of `α`." ]
def smul (c : G) : α ≃ᵐ α 
goal: (MeasurableEquiv.smul c).toEquiv = MulAction.toPerm c

full_name: MeasurableEquiv.vadd_apply
code: /-- If a group `G` acts on `α` by measurable maps, then each element `c : G` defines a measurable
automorphism of `α`. -/
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
      "If an additive group `G` acts on `α` by measurable maps, then each element `c : G`
      defines a measurable automorphism of `α`." ]
def smul (c : G) : α ≃ᵐ α 
goal: ⇑(MeasurableEquiv.vadd c) = fun x => c +ᵥ x

full_name: MeasurableEquiv.smul
code: /-- If a group `G` acts on `α` by measurable maps, then each element `c : G` defines a measurable
automorphism of `α`. -/
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
      "If an additive group `G` acts on `α` by measurable maps, then each element `c : G`
      defines a measurable automorphism of `α`." ]
def smul (c : G) : α ≃ᵐ α 
goal: α ≃ᵐ α

full_name: MeasurableEquiv.smul_apply
code: /-- If a group `G` acts on `α` by measurable maps, then each element `c : G` defines a measurable
automorphism of `α`. -/
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
      "If an additive group `G` acts on `α` by measurable maps, then each element `c : G`
      defines a measurable automorphism of `α`." ]
def smul (c : G) : α ≃ᵐ α 
goal: ⇑(MeasurableEquiv.smul c) = fun x => c • x

full_name: MeasurableEquiv.vadd
code: /-- If a group `G` acts on `α` by measurable maps, then each element `c : G` defines a measurable
automorphism of `α`. -/
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
      "If an additive group `G` acts on `α` by measurable maps, then each element `c : G`
      defines a measurable automorphism of `α`." ]
def smul (c : G) : α ≃ᵐ α 
goal: α ≃ᵐ α

full_name: MeasurableEquiv.vadd_toEquiv
code: /-- If a group `G` acts on `α` by measurable maps, then each element `c : G` defines a measurable
automorphism of `α`. -/
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
      "If an additive group `G` acts on `α` by measurable maps, then each element `c : G`
      defines a measurable automorphism of `α`." ]
def smul (c : G) : α ≃ᵐ α 
goal: (MeasurableEquiv.vadd c).toEquiv = AddAction.toPerm c

full_name: measurableEmbedding_const_smul
code: @[to_additive]
theorem _root_.measurableEmbedding_const_smul (c : G) : MeasurableEmbedding (c • · : α → α) 
goal: MeasurableEmbedding fun x => c • x

full_name: measurableEmbedding_const_vadd
code: @[to_additive]
theorem _root_.measurableEmbedding_const_smul (c : G) : MeasurableEmbedding (c • · : α → α) 
goal: MeasurableEmbedding fun x => c +ᵥ x

full_name: MeasurableEquiv.symm_smul
code: @[to_additive (attr := simp)]
theorem symm_smul (c : G) : (smul c : α ≃ᵐ α).symm = smul c⁻¹ 
goal: (MeasurableEquiv.smul c).symm = MeasurableEquiv.smul c⁻¹

full_name: MeasurableEquiv.symm_vadd
code: @[to_additive (attr := simp)]
theorem symm_smul (c : G) : (smul c : α ≃ᵐ α).symm = smul c⁻¹ 
goal: (MeasurableEquiv.vadd c).symm = MeasurableEquiv.vadd (-c)

full_name: MeasurableEquiv.smul₀
code: /-- If a group with zero `G₀` acts on `α` by measurable maps, then each nonzero element `c : G₀`
defines a measurable automorphism of `α` -/
def smul₀ (c : G₀) (hc : c ≠ 0) : α ≃ᵐ α 
goal: α ≃ᵐ α

full_name: MeasurableEquiv.coe_smul₀
code: @[simp] lemma coe_smul₀ {c : G₀} (hc : c ≠ 0) : ⇑(smul₀ c hc : α ≃ᵐ α) = (c • ·) 
goal: ⇑(MeasurableEquiv.smul₀ c hc) = fun x => c • x

full_name: MeasurableEquiv.symm_smul₀
code: @[simp]
theorem symm_smul₀ {c : G₀} (hc : c ≠ 0) :
    (smul₀ c hc : α ≃ᵐ α).symm = smul₀ c⁻¹ (inv_ne_zero hc) 
goal: (MeasurableEquiv.smul₀ c hc).symm = MeasurableEquiv.smul₀ c⁻¹ ⋯

full_name: measurableEmbedding_const_smul₀
code: theorem _root_.measurableEmbedding_const_smul₀ {c : G₀} (hc : c ≠ 0) :
    MeasurableEmbedding (c • · : α → α) 
goal: MeasurableEmbedding fun x => c • x

full_name: MeasurableEquiv.addLeft
code: /-- If `G` is a group with measurable multiplication, then left multiplication by `g : G` is a
measurable automorphism of `G`. -/
@[to_additive
      "If `G` is an additive group with measurable addition, then addition of `g : G`
      on the left is a measurable automorphism of `G`."]
def mulLeft (g : G) : G ≃ᵐ G 
goal: G ≃ᵐ G

full_name: MeasurableEquiv.mulLeft
code: /-- If `G` is a group with measurable multiplication, then left multiplication by `g : G` is a
measurable automorphism of `G`. -/
@[to_additive
      "If `G` is an additive group with measurable addition, then addition of `g : G`
      on the left is a measurable automorphism of `G`."]
def mulLeft (g : G) : G ≃ᵐ G 
goal: G ≃ᵐ G

full_name: MeasurableEquiv.coe_mulLeft
code: @[to_additive (attr := simp)]
theorem coe_mulLeft (g : G) : ⇑(mulLeft g) = (g * ·) 
goal: ⇑(MeasurableEquiv.mulLeft g) = fun x => g * x

full_name: MeasurableEquiv.coe_addLeft
code: @[to_additive (attr := simp)]
theorem coe_mulLeft (g : G) : ⇑(mulLeft g) = (g * ·) 
goal: ⇑(MeasurableEquiv.addLeft g) = fun x => g + x

full_name: MeasurableEquiv.symm_mulLeft
code: @[to_additive (attr := simp)]
theorem symm_mulLeft (g : G) : (mulLeft g).symm = mulLeft g⁻¹ 
goal: (MeasurableEquiv.mulLeft g).symm = MeasurableEquiv.mulLeft g⁻¹

full_name: MeasurableEquiv.symm_addLeft
code: @[to_additive (attr := simp)]
theorem symm_mulLeft (g : G) : (mulLeft g).symm = mulLeft g⁻¹ 
goal: (MeasurableEquiv.addLeft g).symm = MeasurableEquiv.addLeft (-g)

full_name: MeasurableEquiv.toEquiv_addLeft
code: @[to_additive (attr := simp)]
theorem toEquiv_mulLeft (g : G) : (mulLeft g).toEquiv = Equiv.mulLeft g 
goal: (MeasurableEquiv.addLeft g).toEquiv = Equiv.addLeft g

full_name: MeasurableEquiv.toEquiv_mulLeft
code: @[to_additive (attr := simp)]
theorem toEquiv_mulLeft (g : G) : (mulLeft g).toEquiv = Equiv.mulLeft g 
goal: (MeasurableEquiv.mulLeft g).toEquiv = Equiv.mulLeft g

full_name: measurableEmbedding_mulLeft
code: @[to_additive]
theorem _root_.measurableEmbedding_mulLeft (g : G) : MeasurableEmbedding (g * ·) 
goal: MeasurableEmbedding fun x => g * x

full_name: measurableEmbedding_addLeft
code: @[to_additive]
theorem _root_.measurableEmbedding_mulLeft (g : G) : MeasurableEmbedding (g * ·) 
goal: MeasurableEmbedding fun x => g + x

full_name: MeasurableEquiv.mulRight
code: /-- If `G` is a group with measurable multiplication, then right multiplication by `g : G` is a
measurable automorphism of `G`. -/
@[to_additive
      "If `G` is an additive group with measurable addition, then addition of `g : G`
      on the right is a measurable automorphism of `G`."]
def mulRight (g : G) : G ≃ᵐ G 
goal: G ≃ᵐ G

full_name: MeasurableEquiv.addRight
code: /-- If `G` is a group with measurable multiplication, then right multiplication by `g : G` is a
measurable automorphism of `G`. -/
@[to_additive
      "If `G` is an additive group with measurable addition, then addition of `g : G`
      on the right is a measurable automorphism of `G`."]
def mulRight (g : G) : G ≃ᵐ G 
goal: G ≃ᵐ G

full_name: measurableEmbedding_mulRight
code: @[to_additive]
theorem _root_.measurableEmbedding_mulRight (g : G) : MeasurableEmbedding fun x => x * g 
goal: MeasurableEmbedding fun x => x * g

full_name: measurableEmbedding_addRight
code: @[to_additive]
theorem _root_.measurableEmbedding_mulRight (g : G) : MeasurableEmbedding fun x => x * g 
goal: MeasurableEmbedding fun x => x + g

full_name: MeasurableEquiv.coe_mulRight
code: @[to_additive (attr := simp)]
theorem coe_mulRight (g : G) : ⇑(mulRight g) = fun x => x * g 
goal: ⇑(MeasurableEquiv.mulRight g) = fun x => x * g

full_name: MeasurableEquiv.coe_addRight
code: @[to_additive (attr := simp)]
theorem coe_mulRight (g : G) : ⇑(mulRight g) = fun x => x * g 
goal: ⇑(MeasurableEquiv.addRight g) = fun x => x + g

full_name: MeasurableEquiv.symm_mulRight
code: @[to_additive (attr := simp)]
theorem symm_mulRight (g : G) : (mulRight g).symm = mulRight g⁻¹ 
goal: (MeasurableEquiv.mulRight g).symm = MeasurableEquiv.mulRight g⁻¹

full_name: MeasurableEquiv.symm_addRight
code: @[to_additive (attr := simp)]
theorem symm_mulRight (g : G) : (mulRight g).symm = mulRight g⁻¹ 
goal: (MeasurableEquiv.addRight g).symm = MeasurableEquiv.addRight (-g)

full_name: MeasurableEquiv.toEquiv_addRight
code: @[to_additive (attr := simp)]
theorem toEquiv_mulRight (g : G) : (mulRight g).toEquiv = Equiv.mulRight g 
goal: (MeasurableEquiv.addRight g).toEquiv = Equiv.addRight g

full_name: MeasurableEquiv.toEquiv_mulRight
code: @[to_additive (attr := simp)]
theorem toEquiv_mulRight (g : G) : (mulRight g).toEquiv = Equiv.mulRight g 
goal: (MeasurableEquiv.mulRight g).toEquiv = Equiv.mulRight g

full_name: MeasurableEquiv.mulLeft₀
code: /-- If `G₀` is a group with zero with measurable multiplication, then left multiplication by a
nonzero element `g : G₀` is a measurable automorphism of `G₀`. -/
def mulLeft₀ (g : G₀) (hg : g ≠ 0) : G₀ ≃ᵐ G₀ 
goal: G₀ ≃ᵐ G₀

full_name: measurableEmbedding_mulLeft₀
code: theorem _root_.measurableEmbedding_mulLeft₀ {g : G₀} (hg : g ≠ 0) :
    MeasurableEmbedding (g * ·) 
goal: MeasurableEmbedding fun x => g * x

full_name: MeasurableEquiv.coe_mulLeft₀
code: @[simp] lemma coe_mulLeft₀ {g : G₀} (hg : g ≠ 0) : ⇑(mulLeft₀ g hg) = (g * ·) 
goal: ⇑(MeasurableEquiv.mulLeft₀ g hg) = fun x => g * x

full_name: MeasurableEquiv.symm_mulLeft₀
code: @[simp]
theorem symm_mulLeft₀ {g : G₀} (hg : g ≠ 0) :
    (mulLeft₀ g hg).symm = mulLeft₀ g⁻¹ (inv_ne_zero hg) 
goal: (MeasurableEquiv.mulLeft₀ g hg).symm = MeasurableEquiv.mulLeft₀ g⁻¹ ⋯

full_name: MeasurableEquiv.toEquiv_mulLeft₀
code: @[simp]
theorem toEquiv_mulLeft₀ {g : G₀} (hg : g ≠ 0) : (mulLeft₀ g hg).toEquiv = Equiv.mulLeft₀ g hg 
goal: (MeasurableEquiv.mulLeft₀ g hg).toEquiv = Equiv.mulLeft₀ g hg

full_name: MeasurableEquiv.mulRight₀
code: /-- If `G₀` is a group with zero with measurable multiplication, then right multiplication by a
nonzero element `g : G₀` is a measurable automorphism of `G₀`. -/
def mulRight₀ (g : G₀) (hg : g ≠ 0) : G₀ ≃ᵐ G₀ 
goal: G₀ ≃ᵐ G₀

full_name: measurableEmbedding_mulRight₀
code: theorem _root_.measurableEmbedding_mulRight₀ {g : G₀} (hg : g ≠ 0) :
    MeasurableEmbedding fun x => x * g 
goal: MeasurableEmbedding fun x => x * g

full_name: MeasurableEquiv.coe_mulRight₀
code: @[simp]
theorem coe_mulRight₀ {g : G₀} (hg : g ≠ 0) : ⇑(mulRight₀ g hg) = fun x => x * g 
goal: ⇑(MeasurableEquiv.mulRight₀ g hg) = fun x => x * g

full_name: MeasurableEquiv.symm_mulRight₀
code: @[simp]
theorem symm_mulRight₀ {g : G₀} (hg : g ≠ 0) :
    (mulRight₀ g hg).symm = mulRight₀ g⁻¹ (inv_ne_zero hg) 
goal: (MeasurableEquiv.mulRight₀ g hg).symm = MeasurableEquiv.mulRight₀ g⁻¹ ⋯

full_name: MeasurableEquiv.toEquiv_mulRight₀
code: @[simp]
theorem toEquiv_mulRight₀ {g : G₀} (hg : g ≠ 0) : (mulRight₀ g hg).toEquiv = Equiv.mulRight₀ g hg 
goal: (MeasurableEquiv.mulRight₀ g hg).toEquiv = Equiv.mulRight₀ g hg

full_name: MeasurableEquiv.inv_toEquiv
code: /-- Inversion as a measurable automorphism of a group or group with zero. -/
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
    "Negation as a measurable automorphism of an additive group."]
def inv (G) [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] : G ≃ᵐ G 
goal: (MeasurableEquiv.inv G).toEquiv = Equiv.inv G

full_name: MeasurableEquiv.neg_apply
code: /-- Inversion as a measurable automorphism of a group or group with zero. -/
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
    "Negation as a measurable automorphism of an additive group."]
def inv (G) [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] : G ≃ᵐ G 
goal: ⇑(MeasurableEquiv.neg G) = Neg.neg

full_name: MeasurableEquiv.inv
code: /-- Inversion as a measurable automorphism of a group or group with zero. -/
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
    "Negation as a measurable automorphism of an additive group."]
def inv (G) [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] : G ≃ᵐ G 
goal: G ≃ᵐ G

full_name: MeasurableEquiv.neg_toEquiv
code: /-- Inversion as a measurable automorphism of a group or group with zero. -/
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
    "Negation as a measurable automorphism of an additive group."]
def inv (G) [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] : G ≃ᵐ G 
goal: (MeasurableEquiv.neg G).toEquiv = Equiv.neg G

full_name: MeasurableEquiv.neg
code: /-- Inversion as a measurable automorphism of a group or group with zero. -/
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
    "Negation as a measurable automorphism of an additive group."]
def inv (G) [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] : G ≃ᵐ G 
goal: G ≃ᵐ G

full_name: MeasurableEquiv.inv_apply
code: /-- Inversion as a measurable automorphism of a group or group with zero. -/
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
    "Negation as a measurable automorphism of an additive group."]
def inv (G) [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] : G ≃ᵐ G 
goal: ⇑(MeasurableEquiv.inv G) = Inv.inv

full_name: MeasurableEquiv.symm_inv
code: @[to_additive (attr := simp)]
theorem symm_inv {G} [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] :
    (inv G).symm = inv G 
goal: (MeasurableEquiv.inv G).symm = MeasurableEquiv.inv G

full_name: MeasurableEquiv.symm_neg
code: @[to_additive (attr := simp)]
theorem symm_inv {G} [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] :
    (inv G).symm = inv G 
goal: (MeasurableEquiv.neg G).symm = MeasurableEquiv.neg G

full_name: MeasurableEquiv.divRight
code: /-- `equiv.divRight` as a `MeasurableEquiv`. -/
@[to_additive " `equiv.subRight` as a `MeasurableEquiv` "]
def divRight [MeasurableMul G] (g : G) : G ≃ᵐ G 
goal: G ≃ᵐ G

full_name: MeasurableEquiv.subRight
code: /-- `equiv.divRight` as a `MeasurableEquiv`. -/
@[to_additive " `equiv.subRight` as a `MeasurableEquiv` "]
def divRight [MeasurableMul G] (g : G) : G ≃ᵐ G 
goal: G ≃ᵐ G

full_name: MeasurableEquiv.divLeft
code: /-- `equiv.divLeft` as a `MeasurableEquiv` -/
@[to_additive " `equiv.subLeft` as a `MeasurableEquiv` "]
def divLeft [MeasurableMul G] [MeasurableInv G] (g : G) : G ≃ᵐ G 
goal: G ≃ᵐ G

full_name: MeasurableEquiv.subLeft
code: /-- `equiv.divLeft` as a `MeasurableEquiv` -/
@[to_additive " `equiv.subLeft` as a `MeasurableEquiv` "]
def divLeft [MeasurableMul G] [MeasurableInv G] (g : G) : G ≃ᵐ G 
goal: G ≃ᵐ G