-- Mathlib/SetTheory/Cardinal/Arithmetic.lean
theorem mul_eq_self {c : Cardinal} (h : ℵ₀ ≤ c) : c * c = c

Proof state:
case intro.intro.intro.refine_2.inr
c✝ : Cardinal.{u_1}
h✝ : ℵ₀ ≤ c✝
c : Cardinal.{u_1}
x✝ : ∀ y < c, Acc (fun x1 x2 => x1 < x2) y
α : Type u_1
IH : ∀ y < #α, ℵ₀ ≤ y → y * y ≤ y
ol : ℵ₀ ≤ #α
r : α → α → Prop
wo : IsWellOrder α r
e : (#α).ord = type r
this✝¹ : LinearOrder α := linearOrderOfSTO r
this✝ : IsWellOrder α fun x1 x2 => x1 < x2
g : α × α → α := fun p => p.1 ⊔ p.2
f : α × α ↪ Ordinal.{u_1} × α × α :=
  { toFun := fun p => ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p), p), inj' := ⋯ }
s : α × α → α × α → Prop :=
  ⇑f ⁻¹'o Prod.Lex (fun x1 x2 => x1 < x2) (Prod.Lex (fun x1 x2 => x1 < x2) fun x1 x2 => x1 < x2)
this : IsWellOrder (α × α) s
p : α × α
h : (typein s).toRelEmbedding p < type s
qo : ℵ₀ ≤ (succ ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p))).card
⊢ (typein LT.lt).toRelEmbedding (g p) < (#α).ord

Tactic:
rw [e]

Result:
case intro.intro.intro.refine_2.inr
c✝ : Cardinal.{u_1}
h✝ : ℵ₀ ≤ c✝
c : Cardinal.{u_1}
x✝ : ∀ y < c, Acc (fun x1 x2 => x1 < x2) y
α : Type u_1
IH : ∀ y < #α, ℵ₀ ≤ y → y * y ≤ y
ol : ℵ₀ ≤ #α
r : α → α → Prop
wo : IsWellOrder α r
e : (#α).ord = type r
this✝¹ : LinearOrder α := linearOrderOfSTO r
this✝ : IsWellOrder α fun x1 x2 => x1 < x2
g : α × α → α := fun p => p.1 ⊔ p.2
f : α × α ↪ Ordinal.{u_1} × α × α :=
  { toFun := fun p => ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p), p), inj' := ⋯ }
s : α × α → α × α → Prop :=
  ⇑f ⁻¹'o Prod.Lex (fun x1 x2 => x1 < x2) (Prod.Lex (fun x1 x2 => x1 < x2) fun x1 x2 => x1 < x2)
this : IsWellOrder (α × α) s
p : α × α
h : (typein s).toRelEmbedding p < type s
qo : ℵ₀ ≤ (succ ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p))).card
⊢ (typein LT.lt).toRelEmbedding (g p) < type r

Proof state:
case intro.intro
c✝ : Cardinal.{u_1}
h : ℵ₀ ≤ c✝
c : Cardinal.{u_1}
x✝ : ∀ y < c, Acc (fun x1 x2 => x1 < x2) y
α : Type u_1
IH : ∀ y < #α, ℵ₀ ≤ y → y * y ≤ y
ol : ℵ₀ ≤ #α
r : α → α → Prop
wo : IsWellOrder α r
e : (#α).ord = type r
this✝¹ : LinearOrder α := linearOrderOfSTO r
this✝ : IsWellOrder α fun x1 x2 => x1 < x2
g : α × α → α := fun p => p.1 ⊔ p.2
f : α × α ↪ Ordinal.{u_1} × α × α :=
  { toFun := fun p => ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p), p), inj' := ⋯ }
s : α × α → α × α → Prop :=
  ⇑f ⁻¹'o Prod.Lex (fun x1 x2 => x1 < x2) (Prod.Lex (fun x1 x2 => x1 < x2) fun x1 x2 => x1 < x2)
this : IsWellOrder (α × α) s
⊢ type s ≤ type r

Tactic:
refine le_of_forall_lt fun o h => ?_

Result:
case intro.intro
c✝ : Cardinal.{u_1}
h✝ : ℵ₀ ≤ c✝
c : Cardinal.{u_1}
x✝ : ∀ y < c, Acc (fun x1 x2 => x1 < x2) y
α : Type u_1
IH : ∀ y < #α, ℵ₀ ≤ y → y * y ≤ y
ol : ℵ₀ ≤ #α
r : α → α → Prop
wo : IsWellOrder α r
e : (#α).ord = type r
this✝¹ : LinearOrder α := linearOrderOfSTO r
this✝ : IsWellOrder α fun x1 x2 => x1 < x2
g : α × α → α := fun p => p.1 ⊔ p.2
f : α × α ↪ Ordinal.{u_1} × α × α :=
  { toFun := fun p => ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p), p), inj' := ⋯ }
s : α × α → α × α → Prop :=
  ⇑f ⁻¹'o Prod.Lex (fun x1 x2 => x1 < x2) (Prod.Lex (fun x1 x2 => x1 < x2) fun x1 x2 => x1 < x2)
this : IsWellOrder (α × α) s
o : Ordinal.{u_1}
h : o < type s
⊢ o < type r

Proof state:
case intro.intro
c✝ : Cardinal.{u_1}
h : ℵ₀ ≤ c✝
c : Cardinal.{u_1}
x✝ : ∀ y < c, Acc (fun x1 x2 => x1 < x2) y
α : Type u_1
IH : ∀ y < #α, ℵ₀ ≤ y → y * y ≤ y
ol : ℵ₀ ≤ #α
r : α → α → Prop
wo : IsWellOrder α r
e : (#α).ord = type r
this✝¹ : LinearOrder α := linearOrderOfSTO r
this✝ : IsWellOrder α fun x1 x2 => x1 < x2
g : α × α → α := fun p => p.1 ⊔ p.2
f : α × α ↪ Ordinal.{u_1} × α × α :=
  { toFun := fun p => ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p), p), inj' := ⋯ }
s : α × α → α × α → Prop :=
  ⇑f ⁻¹'o Prod.Lex (fun x1 x2 => x1 < x2) (Prod.Lex (fun x1 x2 => x1 < x2) fun x1 x2 => x1 < x2)
this : IsWellOrder (α × α) s
⊢ #α * #α ≤ #α

Tactic:
suffices type s ≤ type r by exact card_le_card this

Result:
case intro.intro
c✝ : Cardinal.{u_1}
h : ℵ₀ ≤ c✝
c : Cardinal.{u_1}
x✝ : ∀ y < c, Acc (fun x1 x2 => x1 < x2) y
α : Type u_1
IH : ∀ y < #α, ℵ₀ ≤ y → y * y ≤ y
ol : ℵ₀ ≤ #α
r : α → α → Prop
wo : IsWellOrder α r
e : (#α).ord = type r
this✝¹ : LinearOrder α := linearOrderOfSTO r
this✝ : IsWellOrder α fun x1 x2 => x1 < x2
g : α × α → α := fun p => p.1 ⊔ p.2
f : α × α ↪ Ordinal.{u_1} × α × α :=
  { toFun := fun p => ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p), p), inj' := ⋯ }
s : α × α → α × α → Prop :=
  ⇑f ⁻¹'o Prod.Lex (fun x1 x2 => x1 < x2) (Prod.Lex (fun x1 x2 => x1 < x2) fun x1 x2 => x1 < x2)
this : IsWellOrder (α × α) s
⊢ type s ≤ type r

Proof state:
c✝ : Cardinal.{u_1}
h : ℵ₀ ≤ c✝
c : Cardinal.{u_1}
x✝ : ∀ y < c, Acc (fun x1 x2 => x1 < x2) y
α : Type u_1
IH : ∀ y < #α, ℵ₀ ≤ y → y * y ≤ y
ol : ℵ₀ ≤ #α
r : α → α → Prop
wo : IsWellOrder α r
e : (#α).ord = type r
this✝² : LinearOrder α := linearOrderOfSTO r
this✝¹ : IsWellOrder α fun x1 x2 => x1 < x2
g : α × α → α := fun p => p.1 ⊔ p.2
f : α × α ↪ Ordinal.{u_1} × α × α :=
  { toFun := fun p => ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p), p), inj' := ⋯ }
s : α × α → α × α → Prop :=
  ⇑f ⁻¹'o Prod.Lex (fun x1 x2 => x1 < x2) (Prod.Lex (fun x1 x2 => x1 < x2) fun x1 x2 => x1 < x2)
this✝ : IsWellOrder (α × α) s
this : type s ≤ type r
⊢ #α * #α ≤ #α

Tactic:
exact card_le_card this

Result:
no goals

Proof state:
case intro.intro.intro.refine_2.inr
c✝ : Cardinal.{u_1}
h✝ : ℵ₀ ≤ c✝
c : Cardinal.{u_1}
x✝ : ∀ y < c, Acc (fun x1 x2 => x1 < x2) y
α : Type u_1
IH : ∀ y < #α, ℵ₀ ≤ y → y * y ≤ y
ol : ℵ₀ ≤ #α
r : α → α → Prop
wo : IsWellOrder α r
e : (#α).ord = type r
this✝¹ : LinearOrder α := linearOrderOfSTO r
this✝ : IsWellOrder α fun x1 x2 => x1 < x2
g : α × α → α := fun p => p.1 ⊔ p.2
f : α × α ↪ Ordinal.{u_1} × α × α :=
  { toFun := fun p => ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p), p), inj' := ⋯ }
s : α × α → α × α → Prop :=
  ⇑f ⁻¹'o Prod.Lex (fun x1 x2 => x1 < x2) (Prod.Lex (fun x1 x2 => x1 < x2) fun x1 x2 => x1 < x2)
this : IsWellOrder (α × α) s
p : α × α
h : (typein s).toRelEmbedding p < type s
qo : ℵ₀ ≤ (succ ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p))).card
⊢ (succ ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p))).card *
      (succ ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p))).card <
    #α

Tactic:
suffices (succ (typein LT.lt (g p))).card < #α from (IH _ this qo).trans_lt this

Result:
case intro.intro.intro.refine_2.inr
c✝ : Cardinal.{u_1}
h✝ : ℵ₀ ≤ c✝
c : Cardinal.{u_1}
x✝ : ∀ y < c, Acc (fun x1 x2 => x1 < x2) y
α : Type u_1
IH : ∀ y < #α, ℵ₀ ≤ y → y * y ≤ y
ol : ℵ₀ ≤ #α
r : α → α → Prop
wo : IsWellOrder α r
e : (#α).ord = type r
this✝¹ : LinearOrder α := linearOrderOfSTO r
this✝ : IsWellOrder α fun x1 x2 => x1 < x2
g : α × α → α := fun p => p.1 ⊔ p.2
f : α × α ↪ Ordinal.{u_1} × α × α :=
  { toFun := fun p => ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p), p), inj' := ⋯ }
s : α × α → α × α → Prop :=
  ⇑f ⁻¹'o Prod.Lex (fun x1 x2 => x1 < x2) (Prod.Lex (fun x1 x2 => x1 < x2) fun x1 x2 => x1 < x2)
this : IsWellOrder (α × α) s
p : α × α
h : (typein s).toRelEmbedding p < type s
qo : ℵ₀ ≤ (succ ((typein fun x1 x2 => x1 < x2).toRelEmbedding (g p))).card
⊢ (succ ((typein LT.lt).toRelEmbedding (g p))).card < #α