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[Merged by Bors] - chore(AlgebraicGeometry/Restrict): restore two lemmas that were deleted by toolchain bump #40189

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[Merged by Bors] - chore(AlgebraicGeometry/Restrict): restore two lemmas that were deleted by toolchain bump #40189

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14 changes: 14 additions & 0 deletions Mathlib/AlgebraicGeometry/Restrict.lean
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Expand Up @@ -652,6 +652,20 @@ theorem morphismRestrict_appLE {X Y : Scheme.{u}} (f : X ⟶ Y) (U : Y.Opens) (V
rw [Scheme.Hom.appLE, morphismRestrict_app', Scheme.Opens.toScheme_presheaf_map,
Scheme.Hom.appLE_map]

@[reassoc]
theorem morphismRestrict_homOfLE_isoImage_ι_hom
{X : Scheme.{u}} {U V : X.Opens} (e : U ≤ V) (W : Opens V) :
X.homOfLE e ∣_ W ≫ (V.ι.isoImage W).hom =
(U.ι.isoImage (X.homOfLE e ⁻¹ᵁ W)).hom ≫ X.homOfLE (X.ι_image_homOfLE_le_ι_image e W) := by
simp [← cancel_mono (V.ι ''ᵁ W).ι]

@[reassoc]
theorem isoImage_ι_inv_morphismRestrict_homOfLE {X : Scheme.{u}} {U V : X.Opens}
(e : U ≤ V) (W : Opens V) :
(U.ι.isoImage (X.homOfLE e ⁻¹ᵁ W)).inv ≫ X.homOfLE e ∣_ W =
X.homOfLE (X.ι_image_homOfLE_le_ι_image e W) ≫ (V.ι.isoImage W).inv := by
simp [← cancel_mono (V.ι.isoImage W).hom, morphismRestrict_homOfLE_isoImage_ι_hom]

set_option backward.defeqAttrib.useBackward true in
set_option backward.isDefEq.respectTransparency false in
/-- Restricting a morphism onto the image of an open immersion is isomorphic to the base change
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