-- Module: Mathlib.RingTheory.HahnSeries.Lex
Goal:
Γ : Type u_1
R : Type u_2
inst : LinearOrder Γ
inst_1 : LinearOrder R
inst_2 : AddCommGroup R
inst_3 : IsOrderedAddMonoid R
a b : Lex (Prod.{u_1, u_2} Γ (FiniteArchimedeanClass R))
x : Prod.{u_1, u_2} Γ (FiniteArchimedeanClass R)
ao : Γ
ac : FiniteArchimedeanClass R
x_1 : Prod.{u_1, u_2} Γ (FiniteArchimedeanClass R)
bc : FiniteArchimedeanClass R
h :
  LE.le (α := Lex (Prod.{u_1, u_2} Γ (FiniteArchimedeanClass R)))
    ((toLex : (a : Γ × FiniteArchimedeanClass R) → Lex (Γ × FiniteArchimedeanClass R)) (ao, ac) :
      Lex (Prod.{u_1, u_2} Γ (FiniteArchimedeanClass R)))
    ((toLex : (a : Γ × FiniteArchimedeanClass R) → Lex (Γ × FiniteArchimedeanClass R))
        (((ofLex : (a : Lex (Γ × FiniteArchimedeanClass R)) → Γ × FiniteArchimedeanClass R)
              ((toLex : (a : Γ × FiniteArchimedeanClass R) → Lex (Γ × FiniteArchimedeanClass R)) (ao, ac))).1,
          bc) :
      Lex (Prod.{u_1, u_2} Γ (FiniteArchimedeanClass R)))
hle :
  LE.le.{u_2} (α := FiniteArchimedeanClass R)
    ((ofLex : (a : Lex (Γ × FiniteArchimedeanClass R)) → Γ × FiniteArchimedeanClass R)
        ((toLex : (a : Γ × FiniteArchimedeanClass R) → Lex (Γ × FiniteArchimedeanClass R)) (ao, ac))).2
    ((ofLex : (a : Lex (Γ × FiniteArchimedeanClass R)) → Γ × FiniteArchimedeanClass R)
        ((toLex : (a : Γ × FiniteArchimedeanClass R) → Lex (Γ × FiniteArchimedeanClass R))
          (((ofLex : (a : Lex (Γ × FiniteArchimedeanClass R)) → Γ × FiniteArchimedeanClass R)
                ((toLex : (a : Γ × FiniteArchimedeanClass R) → Lex (Γ × FiniteArchimedeanClass R)) (ao, ac))).1,
            bc))).2
⊢ LE.le (α := FiniteArchimedeanClass (Lex R⟦Γ⟧))
    ((fun (x : Lex (Prod.{u_1, u_2} Γ (FiniteArchimedeanClass R))) ↦
        ((FiniteArchimedeanClass.liftOrderHom
                (fun (a : { a : R // a ≠ 0 }) ↦
                  FiniteArchimedeanClass.mk
                    ((toLex : (a : R⟦Γ⟧) → Lex R⟦Γ⟧)
                      ((single ((ofLex : (a : Lex (Γ × FiniteArchimedeanClass R)) → Γ × FiniteArchimedeanClass R) x).1 :
                          (a : R) → R⟦Γ⟧)
                        (↑a : R)))
                    ⋯)
                ⋯ :
              (a : FiniteArchimedeanClass R) → FiniteArchimedeanClass (Lex R⟦Γ⟧))
            ((ofLex : (a : Lex (Γ × FiniteArchimedeanClass R)) → Γ × FiniteArchimedeanClass R) x).2 :
          FiniteArchimedeanClass (Lex R⟦Γ⟧)))
      ((toLex : (a : Γ × FiniteArchimedeanClass R) → Lex (Γ × FiniteArchimedeanClass R)) (ao, ac)))
    ((fun (x : Lex (Prod.{u_1, u_2} Γ (FiniteArchimedeanClass R))) ↦
        ((FiniteArchimedeanClass.liftOrderHom
                (fun (a : { a : R // a ≠ 0 }) ↦
                  FiniteArchimedeanClass.mk
                    ((toLex : (a : R⟦Γ⟧) → Lex R⟦Γ⟧)
                      ((single ((ofLex : (a : Lex (Γ × FiniteArchimedeanClass R)) → Γ × FiniteArchimedeanClass R) x).1 :
                          (a : R) → R⟦Γ⟧)
                        (↑a : R)))
                    ⋯)
                ⋯ :
              (a : FiniteArchimedeanClass R) → FiniteArchimedeanClass (Lex R⟦Γ⟧))
            ((ofLex : (a : Lex (Γ × FiniteArchimedeanClass R)) → Γ × FiniteArchimedeanClass R) x).2 :
          FiniteArchimedeanClass (Lex R⟦Γ⟧)))
      ((toLex : (a : Γ × FiniteArchimedeanClass R) → Lex (Γ × FiniteArchimedeanClass R))
        (((ofLex : (a : Lex (Γ × FiniteArchimedeanClass R)) → Γ × FiniteArchimedeanClass R)
              ((toLex : (a : Γ × FiniteArchimedeanClass R) → Lex (Γ × FiniteArchimedeanClass R)) (ao, ac))).1,
          bc)))

Tactic:
OrderHom.monotone _ hle

Premises used: OrderHom.monotone, Lex, ArchimedeanOrder