Theorem: multiplicity_prime_eq_multiplicity_image_by_factor_orderIso
File: Mathlib/RingTheory/ChainOfDivisors.lean

Proof steps:
state_before: M : Type u_1
inst✝⁵ : CancelCommMonoidWithZero M
N : Type u_2
inst✝⁴ : CancelCommMonoidWithZero N
inst✝³ : UniqueFactorizationMonoid N
inst✝² : UniqueFactorizationMonoid M
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
⊢ multiplicity p m = multiplicity (↑(d ⟨p, ⋯⟩)) n
state_after: M : Type u_1
inst✝⁵ : CancelCommMonoidWithZero M
N : Type u_2
inst✝⁴ : CancelCommMonoidWithZero N
inst✝³ : UniqueFactorizationMonoid N
inst✝² : UniqueFactorizationMonoid M
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity p m
tactic: refine le_antisymm (multiplicity_prime_le_multiplicity_image_by_factor_orderIso hp d) ?_
state_before: M : Type u_1
inst✝⁵ : CancelCommMonoidWithZero M
N : Type u_2
inst✝⁴ : CancelCommMonoidWithZero N
inst✝³ : UniqueFactorizationMonoid N
inst✝² : UniqueFactorizationMonoid M
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity p m
state_after: M : Type u_1
inst✝⁵ : CancelCommMonoidWithZero M
N : Type u_2
inst✝⁴ : CancelCommMonoidWithZero N
inst✝³ : UniqueFactorizationMonoid N
inst✝² : UniqueFactorizationMonoid M
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity (↑(d.symm (d ⟨p, ⋯⟩))) m
tactic: suffices multiplicity (↑(d ⟨p, dvd_of_mem_normalizedFactors hp⟩)) n ≤
      multiplicity (↑(d.symm (d ⟨p, dvd_of_mem_normalizedFactors hp⟩))) m by
    rw [d.symm_apply_apply ⟨p, dvd_of_mem_normalizedFactors hp⟩, Subtype.coe_mk] at this
    exact this
state_before: M : Type u_1
inst✝⁵ : CancelCommMonoidWithZero M
N : Type u_2
inst✝⁴ : CancelCommMonoidWithZero N
inst✝³ : UniqueFactorizationMonoid N
inst✝² : UniqueFactorizationMonoid M
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity (↑(d.symm (d ⟨p, ⋯⟩))) m
state_after: M : Type u_1
inst✝⁵ : CancelCommMonoidWithZero M
N : Type u_2
inst✝⁴ : CancelCommMonoidWithZero N
inst✝³ : UniqueFactorizationMonoid N
inst✝² : UniqueFactorizationMonoid M
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
this : DecidableEq (Associates N) := Classical.decEq (Associates N)
⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity (↑(d.symm (d ⟨p, ⋯⟩))) m
tactic: letI := Classical.decEq (Associates N)
state_before: M : Type u_1
inst✝⁵ : CancelCommMonoidWithZero M
N : Type u_2
inst✝⁴ : CancelCommMonoidWithZero N
inst✝³ : UniqueFactorizationMonoid N
inst✝² : UniqueFactorizationMonoid M
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
this : DecidableEq (Associates N) := Classical.decEq (Associates N)
⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity (↑(d.symm (d ⟨p, ⋯⟩))) m
state_after: no goals
tactic: simpa only [Subtype.coe_eta] using
    multiplicity_prime_le_multiplicity_image_by_factor_orderIso
      (mem_normalizedFactors_factor_orderIso_of_mem_normalizedFactors hn hp d) d.symm