internlm/Lean-Github huggingface dataset:
Theorem: MvPolynomial.coeToMvPowerSeries_uniformContinuous
Proof state before:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Tactic:
let tendsto_zero := ha.tendsto_zero
Proof state after:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
tendsto_zero : Filter.Tendsto a Filter.cofinite (nhds 0) := ha.tendsto_zero
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Theorem: MvPolynomial.coeToMvPowerSeries_uniformContinuous
Proof state before:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
tendsto_zero : Filter.Tendsto a Filter.cofinite (nhds 0) := ha.tendsto_zero
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Tactic:
let hpow := ha.hpow
Proof state after:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
tendsto_zero : Filter.Tendsto a Filter.cofinite (nhds 0) := ha.tendsto_zero
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Theorem: MvPolynomial.coeToMvPowerSeries_uniformContinuous
Proof state before:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
tendsto_zero : Filter.Tendsto a Filter.cofinite (nhds 0) := ha.tendsto_zero
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Tactic:
simp only [Filter.tendsto_def] at tendsto_zero hpow
Proof state after:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : ∀ s ∈ nhds 0, a ⁻¹' s ∈ Filter.cofinite
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Theorem: MvPolynomial.coeToMvPowerSeries_uniformContinuous
Proof state before:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : ∀ s ∈ nhds 0, a ⁻¹' s ∈ Filter.cofinite
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Tactic:
specialize tendsto_zero I hI
Proof state after:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : a ⁻¹' ↑I ∈ Filter.cofinite
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Theorem: MvPolynomial.coeToMvPowerSeries_uniformContinuous
Proof state before:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : a ⁻¹' ↑I ∈ Filter.cofinite
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Tactic:
simp only [Filter.mem_cofinite] at tendsto_zero
Proof state after:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Theorem: MvPolynomial.coeToMvPowerSeries_uniformContinuous
Proof state before:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Tactic:
let hpow' := fun s ↦ hpow s hI
Proof state after:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ↑I ∈ Filter.map (fun n => a s ^ n) Filter.atTop := fun s => hpow s hI
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Theorem: MvPolynomial.coeToMvPowerSeries_uniformContinuous
Proof state before:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ↑I ∈ Filter.map (fun n => a s ^ n) Filter.atTop := fun s => hpow s hI
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Tactic:
simp only [Filter.mem_map, Filter.mem_atTop_sets, ge_iff_le, Set.mem_preimage, SetLike.mem_coe] at hpow'
Proof state after:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Theorem: MvPolynomial.coeToMvPowerSeries_uniformContinuous
Proof state before:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Tactic:
let n : σ → ℕ := fun s ↦ sInf {n : ℕ | (a s) ^ n.succ ∈ I}
Proof state after:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
n : σ → ℕ := fun s => sInf {n | a s ^ n.succ ∈ I}
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Theorem: MvPolynomial.coeToMvPowerSeries_uniformContinuous
Proof state before:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
n : σ → ℕ := fun s => sInf {n | a s ^ n.succ ∈ I}
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Tactic:
have hn_ne : ∀ s, Set.Nonempty {n : ℕ | (a s) ^ n.succ ∈ I} := fun s ↦ by
  rcases hpow' s with ⟨n, hn⟩
  use n
  simp only [Set.mem_setOf_eq]
  refine hn n.succ (Nat.le_succ n)
Proof state after:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
n : σ → ℕ := fun s => sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Theorem: MvPolynomial.coeToMvPowerSeries_uniformContinuous
Proof state before:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
n : σ → ℕ := fun s => sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Tactic:
have hn : Set.Finite (n.support) := by
  apply @Finite.Set.subset  _ _ _ tendsto_zero
  intro s
  simp only [Function.mem_support, ne_eq, Set.mem_compl_iff, Set.mem_preimage, SetLike.mem_coe, not_imp_comm, not_not]
  simp only [imp_or, n, Nat.sInf_eq_zero, Set.mem_setOf_eq, zero_add, pow_one, imp_self, true_or]
Proof state after:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
n : σ → ℕ := fun s => sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Theorem: MvPolynomial.coeToMvPowerSeries_uniformContinuous
Proof state before:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
n : σ → ℕ := fun s => sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Tactic:
let n₀ : σ →₀ ℕ := {
  toFun := n
  support := hn.toFinset
  mem_support_toFun := fun (s : σ) ↦  by simp }
Proof state after:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
n : σ → ℕ := fun s => sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
n₀ : σ →₀ ℕ := { support := hn.toFinset, toFun := n, mem_support_toFun := ⋯ }
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Theorem: MvPolynomial.coeToMvPowerSeries_uniformContinuous
Proof state before:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
n : σ → ℕ := fun s => sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
n₀ : σ →₀ ℕ := { support := hn.toFinset, toFun := n, mem_support_toFun := ⋯ }
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Tactic:
let D := Set.Iic n₀
Proof state after:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
n : σ → ℕ := fun s => sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
n₀ : σ →₀ ℕ := { support := hn.toFinset, toFun := n, mem_support_toFun := ⋯ }
D : Set (σ →₀ ℕ) := Set.Iic n₀
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Theorem: MvPolynomial.coeToMvPowerSeries_uniformContinuous
Proof state before:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
n : σ → ℕ := fun s => sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
n₀ : σ →₀ ℕ := { support := hn.toFinset, toFun := n, mem_support_toFun := ⋯ }
D : Set (σ →₀ ℕ) := Set.Iic n₀
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Tactic:
have hD : Set.Finite D := Set.finite_Iic _
Proof state after:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
n : σ → ℕ := fun s => sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
n₀ : σ →₀ ℕ := { support := hn.toFinset, toFun := n, mem_support_toFun := ⋯ }
D : Set (σ →₀ ℕ) := Set.Iic n₀
hD : D.Finite
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Theorem: MvPolynomial.coeToMvPowerSeries_uniformContinuous
Proof state before:
case hf.intro.intro.intro
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
n : σ → ℕ := fun s => sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
n₀ : σ →₀ ℕ := { support := hn.toFinset, toFun := n, mem_support_toFun := ⋯ }
D : Set (σ →₀ ℕ) := Set.Iic n₀
hD : D.Finite
⊢ ∃ t ∈ nhds 0, toMvPowerSeries ⁻¹' t ⊆ eval₂ φ a ⁻¹' u
Tactic:
use Set.iInter (fun (d : D) ↦ { p | φ (p d.val) ∈ I})
Proof state after:
case h
σ : Type u_1
inst✝¹⁰ : DecidableEq σ
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformAddGroup R
inst✝⁶ : TopologicalRing R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : UniformSpace S
inst✝³ : UniformAddGroup S
inst✝² : TopologicalRing S
inst✝¹ : T2Space S
inst✝ : CompleteSpace S
φ : R →+* S
hφ : Continuous ⇑φ
hS : LinearTopology S
a : σ → S
ha : EvalDomain a
u : Set S
I : Ideal S
left✝ : I ∈ LinearTopology.toIdealBasis.sets
hI : ↑I ∈ nhds 0
hI' : ↑I ≤ u
hpow : ∀ (s : σ), IsTopologicallyNilpotent (a s) := ha.hpow
tendsto_zero : (a ⁻¹' ↑I)ᶜ.Finite
hpow' : ∀ (s : σ), ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a s ^ b ∈ I
n : σ → ℕ := fun s => sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
n₀ : σ →₀ ℕ := { support := hn.toFinset, toFun := n, mem_support_toFun := ⋯ }
D : Set (σ →₀ ℕ) := Set.Iic n₀
hD : D.Finite
⊢ ⋂ d, {p | φ (p ↑d) ∈ I} ∈ nhds 0 ∧ toMvPowerSeries ⁻¹' ⋂ d, {p | φ (p ↑d) ∈ I} ⊆ eval₂ φ a ⁻¹' u