Theorem: PadicSeq.norm_mul
File: Mathlib/NumberTheory/Padics/PadicNumbers.lean

Proof steps:
state_before: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : f ≈ 0
⊢ (f * g).norm = f.norm * g.norm
state_after: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : f ≈ 0
hg : f * g ≈ 0
⊢ (f * g).norm = f.norm * g.norm
tactic: have hg : f * g ≈ 0 := mul_equiv_zero' _ hf
state_before: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : f ≈ 0
hg : f * g ≈ 0
⊢ (f * g).norm = f.norm * g.norm
state_after: no goals
tactic: simp only [hf, hg, norm, dif_pos, zero_mul]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : g ≈ 0
⊢ (f * g).norm = f.norm * g.norm
state_after: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf✝ : ¬f ≈ 0
hg : g ≈ 0
hf : f * g ≈ 0
⊢ (f * g).norm = f.norm * g.norm
tactic: have hf : f * g ≈ 0 := mul_equiv_zero _ hg
state_before: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf✝ : ¬f ≈ 0
hg : g ≈ 0
hf : f * g ≈ 0
⊢ (f * g).norm = f.norm * g.norm
state_after: no goals
tactic: simp only [hf, hg, norm, dif_pos, mul_zero]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
⊢ (f * g).norm = f.norm * g.norm
state_after: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
⊢ (if hf : f * g ≈ 0 then 0 else padicNorm p (↑(f * g) (stationaryPoint hf))) =
    (if hf : f ≈ 0 then 0 else padicNorm p (↑f (stationaryPoint hf))) *
      if hf : g ≈ 0 then 0 else padicNorm p (↑g (stationaryPoint hf))
tactic: unfold norm
state_before: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
⊢ (if hf : f * g ≈ 0 then 0 else padicNorm p (↑(f * g) (stationaryPoint hf))) =
    (if hf : f ≈ 0 then 0 else padicNorm p (↑f (stationaryPoint hf))) *
      if hf : g ≈ 0 then 0 else padicNorm p (↑g (stationaryPoint hf))
state_after: case pos
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f * g ≈ 0
⊢ 0 = padicNorm p (↑f (stationaryPoint hf)) * padicNorm p (↑g (stationaryPoint hg))

case neg
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ padicNorm p (↑(f * g) (stationaryPoint hfg)) =
    padicNorm p (↑f (stationaryPoint hf)) * padicNorm p (↑g (stationaryPoint hg))
tactic: split_ifs with hfg
state_before: case neg
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ padicNorm p (↑(f * g) (stationaryPoint hfg)) =
    padicNorm p (↑f (stationaryPoint hf)) * padicNorm p (↑g (stationaryPoint hg))
state_after: case neg
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ padicNorm p (↑(f * g) (max (stationaryPoint hfg) (max ?neg.v2✝ ?neg.v3✝))) =
    padicNorm p (↑f (max ?neg.v1✝ (max (stationaryPoint hf) ?neg.v3✝))) *
      padicNorm p (↑g (max ?neg.v1✝ (max ?neg.v2✝ (stationaryPoint hg))))

case neg.v1
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ ℕ

case neg.v2
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ ℕ

case neg.v1
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ ℕ

case neg.v3
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ ℕ

case neg.v2
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ ℕ

case neg.v3
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ ℕ
tactic: rw [lift_index_left_left hfg, lift_index_left hf, lift_index_right hg]
state_before: case neg
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ padicNorm p (↑(f * g) (max (stationaryPoint hfg) (max ?neg.v2✝ ?neg.v3✝))) =
    padicNorm p (↑f (max ?neg.v1✝ (max (stationaryPoint hf) ?neg.v3✝))) *
      padicNorm p (↑g (max ?neg.v1✝ (max ?neg.v2✝ (stationaryPoint hg))))

case neg.v1
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ ℕ

case neg.v2
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ ℕ

case neg.v1
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ ℕ

case neg.v3
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ ℕ

case neg.v2
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ ℕ

case neg.v3
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ ℕ
state_after: no goals
tactic: apply padicNorm.mul