case a
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : padicNorm p q = 0
⊢ ¬q = 0 → False
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : padicNorm p q = 0
⊢ q = 0
apply by_contradiction
case a
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : padicNorm p q = 0
hq : ¬q = 0
⊢ False
case a
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : padicNorm p q = 0
⊢ ¬q = 0 → False
intro hq
case a
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : (if q = 0 then 0 else ↑p ^ (-padicValRat p q)) = 0
hq : ¬q = 0
⊢ False
case a
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : padicNorm p q = 0
hq : ¬q = 0
⊢ False
unfold padicNorm at h
case a
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : ↑p ^ (-padicValRat p q) = 0
hq : ¬q = 0
⊢ False
case a
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : (if q = 0 then 0 else ↑p ^ (-padicValRat p q)) = 0
hq : ¬q = 0
⊢ False
rw [if_neg hq] at h
case a
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : ↑p ^ (-padicValRat p q) = 0
hq : ¬q = 0
⊢ ¬↑p ^ (-padicValRat p q) = 0
case a
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : ↑p ^ (-padicValRat p q) = 0
hq : ¬q = 0
⊢ False
apply absurd h
case a.ha
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : ↑p ^ (-padicValRat p q) = 0
hq : ¬q = 0
⊢ ↑p ≠ 0
case a
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : ↑p ^ (-padicValRat p q) = 0
hq : ¬q = 0
⊢ ¬↑p ^ (-padicValRat p q) = 0
apply zpow_ne_zero_of_ne_zero
no goals
case a.ha
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : ↑p ^ (-padicValRat p q) = 0
hq : ¬q = 0
⊢ ↑p ≠ 0
exact_mod_cast hp.1.ne_zero