| {"name":"Function.Bijective.HSub.hSub.arg_a0.Bijective_rule","declaration":"theorem Function.Bijective.HSub.hSub.arg_a0.Bijective_rule {X : Type u_2} {Y : Type u_1} [AddGroup Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) : Function.Bijective fun x => f x - y"} | |
| {"name":"Function.Bijective.HSub.hSub.arg_a1.Bijective_rule","declaration":"theorem Function.Bijective.HSub.hSub.arg_a1.Bijective_rule {X : Type u_2} {Y : Type u_1} [AddGroup Y] (y : Y) (f : X → Y) (hf : Function.Bijective f) : Function.Bijective fun x => y - f x"} | |
| {"name":"Function.invFun_comp'","declaration":"theorem Function.invFun_comp' {α : Sort u_1} {β : Sort u_2} [Nonempty α] {f : α → β} (hf : Function.Injective f) {x : α} : Function.invFun f (f x) = x"} | |
| {"name":"Function.Bijective.Equiv.toFun.arg_a0.Bijective_rule","declaration":"theorem Function.Bijective.Equiv.toFun.arg_a0.Bijective_rule {X : Type u_3} {Y : Type u_1} {Z : Type u_2} (f : Y ≃ Z) (g : X → Y) (hf : Function.Bijective g) : Function.Bijective fun x => f (g x)"} | |
| {"name":"Function.Bijective.HMul.hMul.arg_a1.Bijective_rule_group","declaration":"theorem Function.Bijective.HMul.hMul.arg_a1.Bijective_rule_group {X : Type u_2} {Y : Type u_1} [Group Y] (y : Y) (f : X → Y) (hf : Function.Bijective f) : Function.Bijective fun x => y * f x"} | |
| {"name":"Function.Bijective.HMul.hMul.arg_a0.Bijective_rule_group","declaration":"theorem Function.Bijective.HMul.hMul.arg_a0.Bijective_rule_group {X : Type u_2} {Y : Type u_1} [Group Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) : Function.Bijective fun x => f x * y"} | |
| {"name":"Function.Bijective.Neg.neg.arg_a0.Bijective_rule","declaration":"theorem Function.Bijective.Neg.neg.arg_a0.Bijective_rule {X : Type u_2} {Y : Type u_1} [AddGroup Y] (f : X → Y) (hf : Function.Bijective f) : Function.Bijective fun x => -f x"} | |
| {"name":"Function.Bijective.Function.comp.arg_a0.Bijective_rule","declaration":"theorem Function.Bijective.Function.comp.arg_a0.Bijective_rule {X : Type u_3} {Y : Type u_1} {Z : Type u_2} (f : Y → Z) (hf : Function.Bijective f) (g : X → Y) (hg : Function.Bijective g) : Function.Bijective fun x => (f ∘ g) x"} | |
| {"name":"Function.Bijective.HAdd.hAdd.arg_a1.Bijective_rule","declaration":"theorem Function.Bijective.HAdd.hAdd.arg_a1.Bijective_rule {X : Type u_2} {Y : Type u_1} [AddGroup Y] (y : Y) (f : X → Y) (hf : Function.Bijective f) : Function.Bijective fun x => y + f x"} | |
| {"name":"Function.Bijective.Prod.mk.arg_fstsnd.Bijective_rule_simple'","declaration":"theorem Function.Bijective.Prod.mk.arg_fstsnd.Bijective_rule_simple' {X : Type u_1} {Y : Type u_2} : Function.Bijective fun xy => (xy.2, xy.1)"} | |
| {"name":"Function.Bijective.HSMul.hSMul.arg_a1.Bijective_rule_group","declaration":"theorem Function.Bijective.HSMul.hSMul.arg_a1.Bijective_rule_group {X : Type u_3} {Y : Type u_2} {G : Type u_1} [Group G] [MulAction G Y] (g : G) (f : X → Y) (hf : Function.Bijective f) : Function.Bijective fun x => g • f x"} | |
| {"name":"Function.Bijective.comp_rule","declaration":"theorem Function.Bijective.comp_rule {X : Type u_3} {Y : Type u_1} {Z : Type u_2} (f : Y → Z) (g : X → Y) (hf : Function.Bijective f) (hg : Function.Bijective g) : Function.Bijective fun x => f (g x)"} | |
| {"name":"Function.Bijective.HDiv.hDiv.arg_a0.Bijective_rule_field","declaration":"theorem Function.Bijective.HDiv.hDiv.arg_a0.Bijective_rule_field {X : Type u_2} {Y : Type u_1} [Field Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) (hy : y ≠ 0) : Function.Bijective fun x => f x / y"} | |
| {"name":"Function.Bijective.HMul.hMul.arg_a0.Bijective_rule_field","declaration":"theorem Function.Bijective.HMul.hMul.arg_a0.Bijective_rule_field {X : Type u_2} {Y : Type u_1} [Field Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) (hy : y ≠ 0) : Function.Bijective fun x => f x * y"} | |
| {"name":"Function.Bijective.HDiv.hDiv.arg_a0.Bijective_rule_group","declaration":"theorem Function.Bijective.HDiv.hDiv.arg_a0.Bijective_rule_group {X : Type u_2} {Y : Type u_1} [Group Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) : Function.Bijective fun x => f x / y"} | |
| {"name":"Function.Bijective.id_rule","declaration":"theorem Function.Bijective.id_rule {X : Type u_1} : Function.Bijective fun x => x"} | |
| {"name":"Function.Bijective.Equiv.invFun.arg_a0.Bijective_rule","declaration":"theorem Function.Bijective.Equiv.invFun.arg_a0.Bijective_rule {X : Type u_3} {Y : Type u_1} {Z : Type u_2} (f : Y ≃ Z) (g : X → Z) (hf : Function.Bijective g) : Function.Bijective fun x => f.invFun (g x)"} | |
| {"name":"Function.Bijective.Inv.inv.arg_a0.Bijective_rule_field","declaration":"theorem Function.Bijective.Inv.inv.arg_a0.Bijective_rule_field {X : Type u_2} {Y : Type u_1} [Field Y] (f : X → Y) (hf : Function.Bijective f) (hf' : ∀ (x : X), f x ≠ 0) : Function.Bijective fun x => (f x)⁻¹"} | |
| {"name":"Function.Bijective.id.arg_a.Bijective_rule","declaration":"theorem Function.Bijective.id.arg_a.Bijective_rule {X : Type u_1} : Function.Bijective fun x => id x"} | |
| {"name":"Function.Bijective.HVAdd.hVAdd.arg_a1.Bijective_rule_group","declaration":"theorem Function.Bijective.HVAdd.hVAdd.arg_a1.Bijective_rule_group {X : Type u_3} {Y : Type u_2} {G : Type u_1} [AddGroup G] [AddAction G Y] (g : G) (f : X → Y) (hf : Function.Bijective f) : Function.Bijective fun x => g +ᵥ f x"} | |
| {"name":"Function.Bijective.Prod.mk.arg_fstsnd.Bijective_rule_simple","declaration":"theorem Function.Bijective.Prod.mk.arg_fstsnd.Bijective_rule_simple {X : Type u_1} {Y : Type u_2} : Function.Bijective fun xy => (xy.1, xy.2)"} | |
| {"name":"Function.Bijective.Inv.inv.arg_a0.Bijective_rule_group","declaration":"theorem Function.Bijective.Inv.inv.arg_a0.Bijective_rule_group {X : Type u_2} {Y : Type u_1} [Group Y] (f : X → Y) (hf : Function.Bijective f) : Function.Bijective fun x => (f x)⁻¹"} | |
| {"name":"Function.Bijective.HSMul.hSMul.arg_a1.Bijective_rule_field","declaration":"theorem Function.Bijective.HSMul.hSMul.arg_a1.Bijective_rule_field {X : Type u_3} {Y : Type u_2} {R : Type u_1} [Field R] [MulAction R Y] (r : R) (f : X → Y) (hf : Function.Bijective f) (hr : r ≠ 0) : Function.Bijective fun x => r • f x"} | |
| {"name":"Function.Bijective.Prod.mk.arg_fstsnd.Bijective_rule","declaration":"theorem Function.Bijective.Prod.mk.arg_fstsnd.Bijective_rule {X : Type u_5} {X₁ : Type u_1} {X₂ : Type u_3} {Y : Type u_2} {Z : Type u_4} (f : X₁ → Y) (g : X₂ → Z) (p₁ : X → X₁) (p₂ : X → X₂) (hf : Function.Bijective f) (hg : Function.Bijective g) (hp : Function.Bijective fun x => (p₁ x, p₂ x)) : Function.Bijective fun x => (f (p₁ x), g (p₂ x))"} | |
| {"name":"Function.Bijective.HMul.hMul.arg_a1.Bijective_rule_field","declaration":"theorem Function.Bijective.HMul.hMul.arg_a1.Bijective_rule_field {X : Type u_2} {Y : Type u_1} [Field Y] (y : Y) (f : X → Y) (hf : Function.Bijective f) (hy : y ≠ 0) : Function.Bijective fun x => y * f x"} | |
| {"name":"Function.Bijective.HAdd.hAdd.arg_a0.Bijective_rule","declaration":"theorem Function.Bijective.HAdd.hAdd.arg_a0.Bijective_rule {X : Type u_2} {Y : Type u_1} [AddGroup Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) : Function.Bijective fun x => f x + y"} | |