Theorem: padicNormE.norm_rat_le_one
File: Mathlib/NumberTheory/Padics/PadicNumbers.lean

Proof steps:
state_before: p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : n = 0
⊢ ‖↑{ num := n, den := d, den_nz := hn, reduced := hd }‖ ≤ 1
state_after: p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : n = 0
this : { num := n, den := d, den_nz := hn, reduced := hd } = 0
⊢ ‖↑{ num := n, den := d, den_nz := hn, reduced := hd }‖ ≤ 1
tactic: have : (⟨n, d, hn, hd⟩ : ℚ) = 0 := Rat.zero_iff_num_zero.mpr hnz
state_before: p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : n = 0
this : { num := n, den := d, den_nz := hn, reduced := hd } = 0
⊢ ‖↑{ num := n, den := d, den_nz := hn, reduced := hd }‖ ≤ 1
state_after: no goals
tactic: norm_num [this]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
⊢ ‖↑{ num := n, den := d, den_nz := hn, reduced := hd }‖ ≤ 1
state_after: p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ ‖↑{ num := n, den := d, den_nz := hn, reduced := hd }‖ ≤ 1
tactic: have hnz' : (⟨n, d, hn, hd⟩ : ℚ) ≠ 0 := mt Rat.zero_iff_num_zero.1 hnz
state_before: p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ ‖↑{ num := n, den := d, den_nz := hn, reduced := hd }‖ ≤ 1
state_after: p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ ↑(padicNorm p { num := n, den := d, den_nz := hn, reduced := hd }) ≤ 1
tactic: rw [padicNormE.eq_padicNorm]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ ↑(padicNorm p { num := n, den := d, den_nz := hn, reduced := hd }) ≤ 1
state_after: p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ padicNorm p { num := n, den := d, den_nz := hn, reduced := hd } ≤ 1
tactic: norm_cast
state_before: p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ padicNorm p { num := n, den := d, den_nz := hn, reduced := hd } ≤ 1
state_after: p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ (↑p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num)⁻¹ ≤ 1
tactic: rw [padicNorm.eq_zpow_of_nonzero hnz', padicValRat, neg_sub,
        padicValNat.eq_zero_of_not_dvd hq, Nat.cast_zero, zero_sub, zpow_neg, zpow_natCast]
state_before: p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ (↑p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num)⁻¹ ≤ 1
state_after: case ha
p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ 1 ≤ ↑p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num
tactic: apply inv_le_one
state_before: case ha
p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ 1 ≤ ↑p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num
state_after: case ha
p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ 1 ≤ p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num
tactic: norm_cast
state_before: case ha
p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ 1 ≤ p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num
state_after: case ha.h
p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ 0 < p
tactic: apply one_le_pow
state_before: case ha.h
p : ℕ
hp : Fact (Nat.Prime p)
n : ℤ
d : ℕ
hn : d ≠ 0
hd : n.natAbs.Coprime d
hq : ¬p ∣ d
hnz : ¬n = 0
hnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0
⊢ 0 < p
state_after: no goals
tactic: exact hp.1.pos