| {"state":"ι : Type u_1\nG : Type u_2\ninst✝³ : Group G\ninst✝² : ConditionallyCompleteLattice G\ninst✝¹ : CovariantClass G G (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : Nonempty ι\nf : ι → G\nhf : BddAbove (range f)\na : G\n⊢ (⨆ i, f i) / a = ⨆ i, f i / a","srcUpToTactic":"/-\nCopyright (c) 2021 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury G. Kudryashov\n-/\nimport Mathlib.Algebra.Order.Group.OrderIso\nimport Mathlib.Algebra.Order.Monoid.OrderDual\nimport Mathlib.Data.Set.Pointwise.Basic\nimport Mathlib.Order.Bounds.OrderIso\nimport Mathlib.Order.ConditionallyCompleteLattice.Basic\n\n#align_import algebra.bounds from \"leanprover-community/mathlib\"@\"dd71334db81d0bd444af1ee339a29298bef40734\"\n\n/-!\n# Upper/lower bounds in ordered monoids and groups\n\nIn this file we prove a few facts like “`-s` is bounded above iff `s` is bounded below”\n(`bddAbove_neg`).\n-/\n\n\nopen Function Set\n\nopen Pointwise\n\nsection InvNeg\n\nvariable {G : Type*} [Group G] [Preorder G] [CovariantClass G G (· * ·) (· ≤ ·)]\n [CovariantClass G G (swap (· * ·)) (· ≤ ·)] {s : Set G} {a : G}\n\n@[to_additive (attr := simp)]\ntheorem bddAbove_inv : BddAbove s⁻¹ ↔ BddBelow s :=\n (OrderIso.inv G).bddAbove_preimage\n#align bdd_above_inv bddAbove_inv\n#align bdd_above_neg bddAbove_neg\n\n@[to_additive (attr := simp)]\ntheorem bddBelow_inv : BddBelow s⁻¹ ↔ BddAbove s :=\n (OrderIso.inv G).bddBelow_preimage\n#align bdd_below_inv bddBelow_inv\n#align bdd_below_neg bddBelow_neg\n\n@[to_additive]\ntheorem BddAbove.inv (h : BddAbove s) : BddBelow s⁻¹ :=\n bddBelow_inv.2 h\n#align bdd_above.inv BddAbove.inv\n#align bdd_above.neg BddAbove.neg\n\n@[to_additive]\ntheorem BddBelow.inv (h : BddBelow s) : BddAbove s⁻¹ :=\n bddAbove_inv.2 h\n#align bdd_below.inv BddBelow.inv\n#align bdd_below.neg BddBelow.neg\n\n@[to_additive (attr := simp)]\ntheorem isLUB_inv : IsLUB s⁻¹ a ↔ IsGLB s a⁻¹ :=\n (OrderIso.inv G).isLUB_preimage\n#align is_lub_inv isLUB_inv\n#align is_lub_neg isLUB_neg\n\n@[to_additive]\ntheorem isLUB_inv' : IsLUB s⁻¹ a⁻¹ ↔ IsGLB s a :=\n (OrderIso.inv G).isLUB_preimage'\n#align is_lub_inv' isLUB_inv'\n#align is_lub_neg' isLUB_neg'\n\n@[to_additive]\ntheorem IsGLB.inv (h : IsGLB s a) : IsLUB s⁻¹ a⁻¹ :=\n isLUB_inv'.2 h\n#align is_glb.inv IsGLB.inv\n#align is_glb.neg IsGLB.neg\n\n@[to_additive (attr := simp)]\ntheorem isGLB_inv : IsGLB s⁻¹ a ↔ IsLUB s a⁻¹ :=\n (OrderIso.inv G).isGLB_preimage\n#align is_glb_inv isGLB_inv\n#align is_glb_neg isGLB_neg\n\n@[to_additive]\ntheorem isGLB_inv' : IsGLB s⁻¹ a⁻¹ ↔ IsLUB s a :=\n (OrderIso.inv G).isGLB_preimage'\n#align is_glb_inv' isGLB_inv'\n#align is_glb_neg' isGLB_neg'\n\n@[to_additive]\ntheorem IsLUB.inv (h : IsLUB s a) : IsGLB s⁻¹ a⁻¹ :=\n isGLB_inv'.2 h\n#align is_lub.inv IsLUB.inv\n#align is_lub.neg IsLUB.neg\n\nend InvNeg\n\nsection mul_add\n\nvariable {M : Type*} [Mul M] [Preorder M] [CovariantClass M M (· * ·) (· ≤ ·)]\n [CovariantClass M M (swap (· * ·)) (· ≤ ·)]\n\n@[to_additive]\ntheorem mul_mem_upperBounds_mul {s t : Set M} {a b : M} (ha : a ∈ upperBounds s)\n (hb : b ∈ upperBounds t) : a * b ∈ upperBounds (s * t) :=\n forall_image2_iff.2 fun _ hx _ hy => mul_le_mul' (ha hx) (hb hy)\n#align mul_mem_upper_bounds_mul mul_mem_upperBounds_mul\n#align add_mem_upper_bounds_add add_mem_upperBounds_add\n\n@[to_additive]\ntheorem subset_upperBounds_mul (s t : Set M) :\n upperBounds s * upperBounds t ⊆ upperBounds (s * t) :=\n image2_subset_iff.2 fun _ hx _ hy => mul_mem_upperBounds_mul hx hy\n#align subset_upper_bounds_mul subset_upperBounds_mul\n#align subset_upper_bounds_add subset_upperBounds_add\n\n@[to_additive]\ntheorem mul_mem_lowerBounds_mul {s t : Set M} {a b : M} (ha : a ∈ lowerBounds s)\n (hb : b ∈ lowerBounds t) : a * b ∈ lowerBounds (s * t) :=\n mul_mem_upperBounds_mul (M := Mᵒᵈ) ha hb\n#align mul_mem_lower_bounds_mul mul_mem_lowerBounds_mul\n#align add_mem_lower_bounds_add add_mem_lowerBounds_add\n\n@[to_additive]\ntheorem subset_lowerBounds_mul (s t : Set M) :\n lowerBounds s * lowerBounds t ⊆ lowerBounds (s * t) :=\n subset_upperBounds_mul (M := Mᵒᵈ) _ _\n#align subset_lower_bounds_mul subset_lowerBounds_mul\n#align subset_lower_bounds_add subset_lowerBounds_add\n\n@[to_additive]\ntheorem BddAbove.mul {s t : Set M} (hs : BddAbove s) (ht : BddAbove t) : BddAbove (s * t) :=\n (Nonempty.mul hs ht).mono (subset_upperBounds_mul s t)\n#align bdd_above.mul BddAbove.mul\n#align bdd_above.add BddAbove.add\n\n@[to_additive]\ntheorem BddBelow.mul {s t : Set M} (hs : BddBelow s) (ht : BddBelow t) : BddBelow (s * t) :=\n (Nonempty.mul hs ht).mono (subset_lowerBounds_mul s t)\n#align bdd_below.mul BddBelow.mul\n#align bdd_below.add BddBelow.add\n\nend mul_add\n\nsection ConditionallyCompleteLattice\n\nsection Right\n\nvariable {ι G : Type*} [Group G] [ConditionallyCompleteLattice G]\n [CovariantClass G G (Function.swap (· * ·)) (· ≤ ·)] [Nonempty ι] {f : ι → G}\n\n@[to_additive]\ntheorem ciSup_mul (hf : BddAbove (Set.range f)) (a : G) : (⨆ i, f i) * a = ⨆ i, f i * a :=\n (OrderIso.mulRight a).map_ciSup hf\n#align csupr_mul ciSup_mul\n#align csupr_add ciSup_add\n\n@[to_additive]\ntheorem ciSup_div (hf : BddAbove (Set.range f)) (a : G) : (⨆ i, f i) / a = ⨆ i, f i / a := by\n ","nextTactic":"simp only [div_eq_mul_inv, ciSup_mul hf]","declUpToTactic":"@[to_additive]\ntheorem ciSup_div (hf : BddAbove (Set.range f)) (a : G) : (⨆ i, f i) / a = ⨆ i, f i / a := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Bounds.153_0.kwNNY3mkAxBsoBv","decl":"@[to_additive]\ntheorem ciSup_div (hf : BddAbove (Set.range f)) (a : G) : (⨆ i, f i) / a = ⨆ i, f i / a "} | |
| {"state":"ι : Type u_1\nG : Type u_2\ninst✝³ : Group G\ninst✝² : ConditionallyCompleteLattice G\ninst✝¹ : CovariantClass G G (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : Nonempty ι\nf : ι → G\nhf : BddBelow (range f)\na : G\n⊢ (⨅ i, f i) / a = ⨅ i, f i / a","srcUpToTactic":"/-\nCopyright (c) 2021 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury G. Kudryashov\n-/\nimport Mathlib.Algebra.Order.Group.OrderIso\nimport Mathlib.Algebra.Order.Monoid.OrderDual\nimport Mathlib.Data.Set.Pointwise.Basic\nimport Mathlib.Order.Bounds.OrderIso\nimport Mathlib.Order.ConditionallyCompleteLattice.Basic\n\n#align_import algebra.bounds from \"leanprover-community/mathlib\"@\"dd71334db81d0bd444af1ee339a29298bef40734\"\n\n/-!\n# Upper/lower bounds in ordered monoids and groups\n\nIn this file we prove a few facts like “`-s` is bounded above iff `s` is bounded below”\n(`bddAbove_neg`).\n-/\n\n\nopen Function Set\n\nopen Pointwise\n\nsection InvNeg\n\nvariable {G : Type*} [Group G] [Preorder G] [CovariantClass G G (· * ·) (· ≤ ·)]\n [CovariantClass G G (swap (· * ·)) (· ≤ ·)] {s : Set G} {a : G}\n\n@[to_additive (attr := simp)]\ntheorem bddAbove_inv : BddAbove s⁻¹ ↔ BddBelow s :=\n (OrderIso.inv G).bddAbove_preimage\n#align bdd_above_inv bddAbove_inv\n#align bdd_above_neg bddAbove_neg\n\n@[to_additive (attr := simp)]\ntheorem bddBelow_inv : BddBelow s⁻¹ ↔ BddAbove s :=\n (OrderIso.inv G).bddBelow_preimage\n#align bdd_below_inv bddBelow_inv\n#align bdd_below_neg bddBelow_neg\n\n@[to_additive]\ntheorem BddAbove.inv (h : BddAbove s) : BddBelow s⁻¹ :=\n bddBelow_inv.2 h\n#align bdd_above.inv BddAbove.inv\n#align bdd_above.neg BddAbove.neg\n\n@[to_additive]\ntheorem BddBelow.inv (h : BddBelow s) : BddAbove s⁻¹ :=\n bddAbove_inv.2 h\n#align bdd_below.inv BddBelow.inv\n#align bdd_below.neg BddBelow.neg\n\n@[to_additive (attr := simp)]\ntheorem isLUB_inv : IsLUB s⁻¹ a ↔ IsGLB s a⁻¹ :=\n (OrderIso.inv G).isLUB_preimage\n#align is_lub_inv isLUB_inv\n#align is_lub_neg isLUB_neg\n\n@[to_additive]\ntheorem isLUB_inv' : IsLUB s⁻¹ a⁻¹ ↔ IsGLB s a :=\n (OrderIso.inv G).isLUB_preimage'\n#align is_lub_inv' isLUB_inv'\n#align is_lub_neg' isLUB_neg'\n\n@[to_additive]\ntheorem IsGLB.inv (h : IsGLB s a) : IsLUB s⁻¹ a⁻¹ :=\n isLUB_inv'.2 h\n#align is_glb.inv IsGLB.inv\n#align is_glb.neg IsGLB.neg\n\n@[to_additive (attr := simp)]\ntheorem isGLB_inv : IsGLB s⁻¹ a ↔ IsLUB s a⁻¹ :=\n (OrderIso.inv G).isGLB_preimage\n#align is_glb_inv isGLB_inv\n#align is_glb_neg isGLB_neg\n\n@[to_additive]\ntheorem isGLB_inv' : IsGLB s⁻¹ a⁻¹ ↔ IsLUB s a :=\n (OrderIso.inv G).isGLB_preimage'\n#align is_glb_inv' isGLB_inv'\n#align is_glb_neg' isGLB_neg'\n\n@[to_additive]\ntheorem IsLUB.inv (h : IsLUB s a) : IsGLB s⁻¹ a⁻¹ :=\n isGLB_inv'.2 h\n#align is_lub.inv IsLUB.inv\n#align is_lub.neg IsLUB.neg\n\nend InvNeg\n\nsection mul_add\n\nvariable {M : Type*} [Mul M] [Preorder M] [CovariantClass M M (· * ·) (· ≤ ·)]\n [CovariantClass M M (swap (· * ·)) (· ≤ ·)]\n\n@[to_additive]\ntheorem mul_mem_upperBounds_mul {s t : Set M} {a b : M} (ha : a ∈ upperBounds s)\n (hb : b ∈ upperBounds t) : a * b ∈ upperBounds (s * t) :=\n forall_image2_iff.2 fun _ hx _ hy => mul_le_mul' (ha hx) (hb hy)\n#align mul_mem_upper_bounds_mul mul_mem_upperBounds_mul\n#align add_mem_upper_bounds_add add_mem_upperBounds_add\n\n@[to_additive]\ntheorem subset_upperBounds_mul (s t : Set M) :\n upperBounds s * upperBounds t ⊆ upperBounds (s * t) :=\n image2_subset_iff.2 fun _ hx _ hy => mul_mem_upperBounds_mul hx hy\n#align subset_upper_bounds_mul subset_upperBounds_mul\n#align subset_upper_bounds_add subset_upperBounds_add\n\n@[to_additive]\ntheorem mul_mem_lowerBounds_mul {s t : Set M} {a b : M} (ha : a ∈ lowerBounds s)\n (hb : b ∈ lowerBounds t) : a * b ∈ lowerBounds (s * t) :=\n mul_mem_upperBounds_mul (M := Mᵒᵈ) ha hb\n#align mul_mem_lower_bounds_mul mul_mem_lowerBounds_mul\n#align add_mem_lower_bounds_add add_mem_lowerBounds_add\n\n@[to_additive]\ntheorem subset_lowerBounds_mul (s t : Set M) :\n lowerBounds s * lowerBounds t ⊆ lowerBounds (s * t) :=\n subset_upperBounds_mul (M := Mᵒᵈ) _ _\n#align subset_lower_bounds_mul subset_lowerBounds_mul\n#align subset_lower_bounds_add subset_lowerBounds_add\n\n@[to_additive]\ntheorem BddAbove.mul {s t : Set M} (hs : BddAbove s) (ht : BddAbove t) : BddAbove (s * t) :=\n (Nonempty.mul hs ht).mono (subset_upperBounds_mul s t)\n#align bdd_above.mul BddAbove.mul\n#align bdd_above.add BddAbove.add\n\n@[to_additive]\ntheorem BddBelow.mul {s t : Set M} (hs : BddBelow s) (ht : BddBelow t) : BddBelow (s * t) :=\n (Nonempty.mul hs ht).mono (subset_lowerBounds_mul s t)\n#align bdd_below.mul BddBelow.mul\n#align bdd_below.add BddBelow.add\n\nend mul_add\n\nsection ConditionallyCompleteLattice\n\nsection Right\n\nvariable {ι G : Type*} [Group G] [ConditionallyCompleteLattice G]\n [CovariantClass G G (Function.swap (· * ·)) (· ≤ ·)] [Nonempty ι] {f : ι → G}\n\n@[to_additive]\ntheorem ciSup_mul (hf : BddAbove (Set.range f)) (a : G) : (⨆ i, f i) * a = ⨆ i, f i * a :=\n (OrderIso.mulRight a).map_ciSup hf\n#align csupr_mul ciSup_mul\n#align csupr_add ciSup_add\n\n@[to_additive]\ntheorem ciSup_div (hf : BddAbove (Set.range f)) (a : G) : (⨆ i, f i) / a = ⨆ i, f i / a := by\n simp only [div_eq_mul_inv, ciSup_mul hf]\n#align csupr_div ciSup_div\n#align csupr_sub ciSup_sub\n\n@[to_additive]\ntheorem ciInf_mul (hf : BddBelow (Set.range f)) (a : G) : (⨅ i, f i) * a = ⨅ i, f i * a :=\n (OrderIso.mulRight a).map_ciInf hf\n\n@[to_additive]\ntheorem ciInf_div (hf : BddBelow (Set.range f)) (a : G) : (⨅ i, f i) / a = ⨅ i, f i / a := by\n ","nextTactic":"simp only [div_eq_mul_inv, ciInf_mul hf]","declUpToTactic":"@[to_additive]\ntheorem ciInf_div (hf : BddBelow (Set.range f)) (a : G) : (⨅ i, f i) / a = ⨅ i, f i / a := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Bounds.163_0.kwNNY3mkAxBsoBv","decl":"@[to_additive]\ntheorem ciInf_div (hf : BddBelow (Set.range f)) (a : G) : (⨅ i, f i) / a = ⨅ i, f i / a "} | |