Theorem: smoothSheafCommRing.ι_forgetStalk_inv
File: Mathlib/Geometry/Manifold/Sheaf/Smooth.lean

Proof steps:
state_before: 𝕜 : Type u_1
inst✝²¹ : NontriviallyNormedField 𝕜
EM : Type u_2
inst✝²⁰ : NormedAddCommGroup EM
inst✝¹⁹ : NormedSpace 𝕜 EM
HM : Type u_3
inst✝¹⁸ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_5
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst✝¹⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace HM M
N G A A' R : Type u
inst✝¹¹ : TopologicalSpace N
inst✝¹⁰ : ChartedSpace H N
inst✝⁹ : TopologicalSpace G
inst✝⁸ : ChartedSpace H G
inst✝⁷ : TopologicalSpace A
inst✝⁶ : ChartedSpace H A
inst✝⁵ : TopologicalSpace A'
inst✝⁴ : ChartedSpace H' A'
inst✝³ : TopologicalSpace R
inst✝² : ChartedSpace H R
inst✝¹ : CommRing R
inst✝ : SmoothRing I R
x : ↑(TopCat.of M)
U : (OpenNhds x)ᵒᵖ
⊢ colimit.ι ((OpenNhds.inclusion x).op ⋙ (smoothSheaf IM I M R).presheaf) U ≫ (forgetStalk IM I M R x).inv =
    (forget CommRingCat).map (colimit.ι ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf) U)
state_after: 𝕜 : Type u_1
inst✝²¹ : NontriviallyNormedField 𝕜
EM : Type u_2
inst✝²⁰ : NormedAddCommGroup EM
inst✝¹⁹ : NormedSpace 𝕜 EM
HM : Type u_3
inst✝¹⁸ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_5
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst✝¹⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace HM M
N G A A' R : Type u
inst✝¹¹ : TopologicalSpace N
inst✝¹⁰ : ChartedSpace H N
inst✝⁹ : TopologicalSpace G
inst✝⁸ : ChartedSpace H G
inst✝⁷ : TopologicalSpace A
inst✝⁶ : ChartedSpace H A
inst✝⁵ : TopologicalSpace A'
inst✝⁴ : ChartedSpace H' A'
inst✝³ : TopologicalSpace R
inst✝² : ChartedSpace H R
inst✝¹ : CommRing R
inst✝ : SmoothRing I R
x : ↑(TopCat.of M)
U : (OpenNhds x)ᵒᵖ
⊢ ⇑(colimit.ι ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf) U) ≫ (forgetStalk IM I M R x).hom =
    ⇑(colimit.ι ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf) U) ≫ (forgetStalk IM I M R x).hom
tactic: rw [Iso.comp_inv_eq, ← smoothSheafCommRing.ι_forgetStalk_hom, CommRingCat.forget_map]
state_before: 𝕜 : Type u_1
inst✝²¹ : NontriviallyNormedField 𝕜
EM : Type u_2
inst✝²⁰ : NormedAddCommGroup EM
inst✝¹⁹ : NormedSpace 𝕜 EM
HM : Type u_3
inst✝¹⁸ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_5
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst✝¹⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace HM M
N G A A' R : Type u
inst✝¹¹ : TopologicalSpace N
inst✝¹⁰ : ChartedSpace H N
inst✝⁹ : TopologicalSpace G
inst✝⁸ : ChartedSpace H G
inst✝⁷ : TopologicalSpace A
inst✝⁶ : ChartedSpace H A
inst✝⁵ : TopologicalSpace A'
inst✝⁴ : ChartedSpace H' A'
inst✝³ : TopologicalSpace R
inst✝² : ChartedSpace H R
inst✝¹ : CommRing R
inst✝ : SmoothRing I R
x : ↑(TopCat.of M)
U : (OpenNhds x)ᵒᵖ
⊢ ⇑(colimit.ι ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf) U) ≫ (forgetStalk IM I M R x).hom =
    ⇑(colimit.ι ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf) U) ≫ (forgetStalk IM I M R x).hom
state_after: no goals
tactic: simp_rw [Functor.comp_obj, Functor.op_obj]