-- Mathlib/Geometry/Manifold/Sheaf/LocallyRingedSpace.lean
theorem smoothSheafCommRing.isUnit_stalk_iff {x : M}
    (f : (smoothSheafCommRing IM 𝓘(𝕜) M 𝕜).presheaf.stalk x) :
    IsUnit f ↔ f ∉ RingHom.ker (smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x)

Proof state:
case intro
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
S : TopCat.Presheaf CommRingCat (TopCat.of M) := (smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf
U : Opens M
hxU : x ∈ U
f : C^∞⟮IM, ↥U; 𝓘(𝕜, 𝕜), 𝕜⟯
hf : (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x) ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f) ≠ 0
hf' : f ⟨x, hxU⟩ ≠ 0
V : Opens M
hUV : V ≤ U
hV₀f : ∀ y ∈ Set.range (Set.inclusion hUV), f y ≠ 0
hV₀ : IsOpen (Set.range (Set.inclusion hUV))
x₀ : ↑↑V
hxx₀ : Set.inclusion hUV x₀ = ⟨x, hxU⟩
⊢ x ∈ ↑V

Tactic:
convert x₀.2

Result:
case h.e'_5
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
S : TopCat.Presheaf CommRingCat (TopCat.of M) := (smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf
U : Opens M
hxU : x ∈ U
f : C^∞⟮IM, ↥U; 𝓘(𝕜, 𝕜), 𝕜⟯
hf : (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x) ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f) ≠ 0
hf' : f ⟨x, hxU⟩ ≠ 0
V : Opens M
hUV : V ≤ U
hV₀f : ∀ y ∈ Set.range (Set.inclusion hUV), f y ≠ 0
hV₀ : IsOpen (Set.range (Set.inclusion hUV))
x₀ : ↑↑V
hxx₀ : Set.inclusion hUV x₀ = ⟨x, hxU⟩
⊢ x = ↑x₀

Proof state:
case h.e'_2
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
S : TopCat.Presheaf CommRingCat (TopCat.of M) := (smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf
U : Opens M
hxU : x ∈ U
f : C^∞⟮IM, ↥U; 𝓘(𝕜, 𝕜), 𝕜⟯
hf : (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x) ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f) ≠ 0
⊢ f ⟨x, hxU⟩ = (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x) ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f)

Tactic:
exact (smoothSheafCommRing.eval_germ U x hxU f).symm

Result:
no goals

Proof state:
case mpr
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
f : ↑((smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf.stalk x)
S : TopCat.Presheaf CommRingCat (TopCat.of M) := (smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf
hf : (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x) f ≠ 0
⊢ IsUnit f

Tactic:
obtain ⟨U : Opens M, hxU, f : C^∞⟮IM, U; 𝓘(𝕜), 𝕜⟯, rfl⟩ := S.germ_exist x f

Result:
case mpr.intro.intro.intro
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
S : TopCat.Presheaf CommRingCat (TopCat.of M) := (smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf
U : Opens M
hxU : x ∈ U
f : C^∞⟮IM, ↥U; 𝓘(𝕜, 𝕜), 𝕜⟯
hf : (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x) ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f) ≠ 0
⊢ IsUnit ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f)

Proof state:
case h.e'_2.h.h
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
S : TopCat.Presheaf CommRingCat (TopCat.of M) := (smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf
U : Opens M
hxU : x ∈ U
f : C^∞⟮IM, ↥U; 𝓘(𝕜, 𝕜), 𝕜⟯
hf : (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x) ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f) ≠ 0
hf' : f ⟨x, hxU⟩ ≠ 0
V₀ : Set ↥U
hV₀f : ∀ y ∈ V₀, f y ≠ 0
hV₀ : IsOpen V₀
hxV₀ : ⟨x, hxU⟩ ∈ V₀
V : Opens M := { carrier := Subtype.val '' V₀, is_open' := ⋯ }
hUV : V ≤ U
e_1✝ : Set ↥U = Set ↑↑U
y : ↥U
⊢ y ∈ V₀ ↔ y ∈ Subtype.val ⁻¹' (Subtype.val '' V₀)

Tactic:
rw [Set.preimage_image_eq _ Subtype.coe_injective]

Result:
no goals

Proof state:
case h.e'_2.h.e'_6.a.h.h
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
S : TopCat.Presheaf CommRingCat (TopCat.of M) := (smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf
U : Opens M
hxU : x ∈ U
f : C^∞⟮IM, ↥U; 𝓘(𝕜, 𝕜), 𝕜⟯
hf : (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x) ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f) ≠ 0
hf' : f ⟨x, hxU⟩ ≠ 0
V : Opens M
hUV : V ≤ U
hV₀f : ∀ y ∈ Set.range (Set.inclusion hUV), f y ≠ 0
hV₀ : IsOpen (Set.range (Set.inclusion hUV))
hxV₀ : ⟨x, hxU⟩ ∈ Set.range (Set.inclusion hUV)
hxV : x ∈ ↑V
hVf : ∀ (y : ↥V), f (Set.inclusion hUV y) ≠ 0
g : C^∞⟮IM, ↥V; 𝓘(𝕜, 𝕜), 𝕜⟯ := ⟨(⇑f ∘ Set.inclusion hUV)⁻¹, ?mpr.intro.intro.intro.intro.intro.intro.refine_1⟩
y : ↥(Opposite.unop (Opposite.op V))
⊢ ((ContMDiffMap.restrictRingHom IM 𝓘(𝕜, 𝕜) 𝕜 hUV) f) y ≠ 0

Tactic:
exact hVf y

Result:
no goals

Proof state:
case mpr.intro.intro.intro.intro.intro.intro
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
S : TopCat.Presheaf CommRingCat (TopCat.of M) := (smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf
U : Opens M
hxU : x ∈ U
f : C^∞⟮IM, ↥U; 𝓘(𝕜, 𝕜), 𝕜⟯
hf : (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x) ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f) ≠ 0
hf' : f ⟨x, hxU⟩ ≠ 0
V : Opens M
hUV : V ≤ U
hV₀f : ∀ y ∈ Set.range (Set.inclusion hUV), f y ≠ 0
hV₀ : IsOpen (Set.range (Set.inclusion hUV))
hxV₀ : ⟨x, hxU⟩ ∈ Set.range (Set.inclusion hUV)
hxV : x ∈ ↑V
⊢ IsUnit ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f)

Tactic:
have hVf : ∀ y : V, f (Set.inclusion hUV y) ≠ 0 :=
  fun y ↦ hV₀f (Set.inclusion hUV y) (Set.mem_range_self y)

Result:
case mpr.intro.intro.intro.intro.intro.intro
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
S : TopCat.Presheaf CommRingCat (TopCat.of M) := (smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf
U : Opens M
hxU : x ∈ U
f : C^∞⟮IM, ↥U; 𝓘(𝕜, 𝕜), 𝕜⟯
hf : (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x) ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f) ≠ 0
hf' : f ⟨x, hxU⟩ ≠ 0
V : Opens M
hUV : V ≤ U
hV₀f : ∀ y ∈ Set.range (Set.inclusion hUV), f y ≠ 0
hV₀ : IsOpen (Set.range (Set.inclusion hUV))
hxV₀ : ⟨x, hxU⟩ ∈ Set.range (Set.inclusion hUV)
hxV : x ∈ ↑V
hVf : ∀ (y : ↥V), f (Set.inclusion hUV y) ≠ 0
⊢ IsUnit ((CategoryTheory.ConcreteCategory.hom (S.germ U x hxU)) f)