/-- Dedekind's linear independence of characters -/
theorem linearIndependent_monoidHom (G : Type*) [Monoid G] (L : Type*) [CommRing L]
    [NoZeroDivisors L] : LinearIndependent L (M := G → L) (fun f => f : (G →* L) → G → L) := by
  -- Porting note: Some casts are required.
  letI := Classical.decEq (G →* L);
  letI : MulAction L L := DistribMulAction.toMulAction;
  -- We prove linear independence by showing that only the trivial linear combination vanishes.
  exact linearIndependent_iff'.2
    -- To do this, we use `Finset` induction,
    -- Porting note: `False.elim` → `fun h => False.elim <| Finset.not_mem_empty _ h`
    fun s =>
      Finset.induction_on s
        (fun g _hg i h => False.elim <| Finset.not_mem_empty _ h) fun a s has ih g hg =>
        -- Here
        -- * `a` is a new character we will insert into the `Finset` of characters `s`,
        -- * `ih` is the fact that only the trivial linear combination of characters in `s` is zero
        -- * `hg` is the fact that `g` are the coefficients of a linear combination summing to zero
        -- and it remains to prove that `g` vanishes on `insert a s`.
        -- We now make the key calculation:
        -- For any character `i` in the original `Finset`, we have `g i • i = g i • a` as functions
        -- on the monoid `G`.
        have h1 : ∀ i ∈ s, (g i • (i : G → L)) = g i • (a : G → L) := fun i his =>
          funext fun x : G =>
            -- We prove these expressions are equal by showing
            -- the differences of their values on each monoid element `x` is zero
            eq_of_sub_eq_zero <|
            ih (fun j => g j * j x - g j * a x)
              (funext fun y : G => calc
                -- After that, it's just a chase scene.
                (∑ i in s, ((g i * i x - g i * a x) • (i : G → L))) y =
                    ∑ i in s, (g i * i x - g i * a x) * i y :=
                  Finset.sum_apply ..
                _ = ∑ i in s, (g i * i x * i y - g i * a x * i y) :=
                  Finset.sum_congr rfl fun _ _ => sub_mul ..
                _ = (∑ i in s, g i * i x * i y) - ∑ i in s, g i * a x * i y :=
                  Finset.sum_sub_distrib
                _ =
                    (g a * a x * a y + ∑ i in s, g i * i x * i y) -
                      (g a * a x * a y + ∑ i in s, g i * a x * i y) :=
                  by rw [add_sub_add_left_eq_sub]
                _ =
                    (∑ i in insert a s, g i * i x * i y) -
                      ∑ i in insert a s, g i * a x * i y :=
                  by rw [Finset.sum_insert has, Finset.sum_insert has]
                _ =
                    (∑ i in insert a s, g i * i (x * y)) -
                      ∑ i in insert a s, a x * (g i * i y) :=
                  congr
                    (congr_arg Sub.sub
                      (Finset.sum_congr rfl fun i _ => by rw [i.map_mul, mul_assoc]))
                    (Finset.sum_congr rfl fun _ _ => by rw [mul_assoc, mul_left_comm])
                _ =
                    (∑ i in insert a s, (g i • (i : G → L))) (x * y) -
                      a x * (∑ i in insert a s, (g i • (i : G → L))) y :=
                  by rw [Finset.sum_apply, Finset.sum_apply, Finset.mul_sum]; rfl
                _ = 0 - a x * 0 := by rw [hg]; rfl
                _ = 0 := by rw [mul_zero, sub_zero]
                )
              i his
        -- On the other hand, since `a` is not already in `s`, for any character `i ∈ s`
        -- there is some element of the monoid on which it differs from `a`.
        have h2 : ∀ i : G →* L, i ∈ s → ∃ y, i y ≠ a y := fun i his =>
          Classical.by_contradiction fun h =>
            have hia : i = a := MonoidHom.ext fun y =>
              Classical.by_contradiction fun hy => h ⟨y, hy⟩
            has <| hia ▸ his
        -- From these two facts we deduce that `g` actually vanishes on `s`,
        have h3 : ∀ i ∈ s, g i = 0 := fun i his =>
          let ⟨y, hy⟩ := h2 i his
          have h : g i • i y = g i • a y := congr_fun (h1 i his) y
          Or.resolve_right (mul_eq_zero.1 <| by rw [mul_sub, sub_eq_zero]; exact h)
            (sub_ne_zero_of_ne hy)
        -- And so, using the fact that the linear combination over `s` and over `insert a s` both
        -- vanish, we deduce that `g a = 0`.
        have h4 : g a = 0 :=
          calc
            g a = g a * 1 := (mul_one _).symm
            _ = (g a • (a : G → L)) 1 := by rw [← a.map_one]; rfl
            _ = (∑ i in insert a s, (g i • (i : G → L))) 1 := by
              

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁹ : Ring R
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup M'
inst✝⁶ : AddCommGroup M''
inst✝⁵ : Module R M
inst✝⁴ : Module R M'
inst✝³ : Module R M''
a✝ b : R
x y : M
G : Type u_8
inst✝² : Monoid G
L : Type u_9
inst✝¹ : CommRing L
inst✝ : NoZeroDivisors L
this✝ : DecidableEq (G →* L) := Classical.decEq (G →* L)
this : MulAction L L := DistribMulAction.toMulAction
s✝ : Finset (G →* L)
a : G →* L
s : Finset (G →* L)
has : a ∉ s
ih : ∀ (g : (G →* L) → L), ∑ i in s, g i • ⇑i = 0 → ∀ i ∈ s, g i = 0
g : (G →* L) → L
hg : ∑ i in insert a s, g i • ⇑i = 0
h1 : ∀ i ∈ s, g i • ⇑i = g i • ⇑a
h2 : ∀ i ∈ s, ∃ y, i y ≠ a y
h3 : ∀ i ∈ s, g i = 0
⊢ (g a • ⇑a) 1 = Finset.sum (insert a s) (fun i => g i • ⇑i) 1

Next tactic:
rw [Finset.sum_eq_single a]