File: Mathlib/LinearAlgebra/AffineSpace/Ordered.lean

full_name: lineMap_mono_left
code: theorem lineMap_mono_left (ha : a ≤ a') (hr : r ≤ 1) : lineMap a b r ≤ lineMap a' b r 
goal: (AffineMap.lineMap a b) r ≤ (AffineMap.lineMap a' b) r

full_name: lineMap_strict_mono_left
code: theorem lineMap_strict_mono_left (ha : a < a') (hr : r < 1) : lineMap a b r < lineMap a' b r 
goal: (AffineMap.lineMap a b) r < (AffineMap.lineMap a' b) r

full_name: lineMap_mono_right
code: theorem lineMap_mono_right (hb : b ≤ b') (hr : 0 ≤ r) : lineMap a b r ≤ lineMap a b' r 
goal: (AffineMap.lineMap a b) r ≤ (AffineMap.lineMap a b') r

full_name: lineMap_strict_mono_right
code: theorem lineMap_strict_mono_right (hb : b < b') (hr : 0 < r) : lineMap a b r < lineMap a b' r 
goal: (AffineMap.lineMap a b) r < (AffineMap.lineMap a b') r

full_name: lineMap_mono_endpoints
code: theorem lineMap_mono_endpoints (ha : a ≤ a') (hb : b ≤ b') (h₀ : 0 ≤ r) (h₁ : r ≤ 1) :
    lineMap a b r ≤ lineMap a' b' r 
goal: (AffineMap.lineMap a b) r ≤ (AffineMap.lineMap a' b') r

full_name: lineMap_strict_mono_endpoints
code: theorem lineMap_strict_mono_endpoints (ha : a < a') (hb : b < b') (h₀ : 0 ≤ r) (h₁ : r ≤ 1) :
    lineMap a b r < lineMap a' b' r 
goal: (AffineMap.lineMap a b) r < (AffineMap.lineMap a' b') r

full_name: lineMap_lt_lineMap_iff_of_lt
code: theorem lineMap_lt_lineMap_iff_of_lt (h : r < r') : lineMap a b r < lineMap a b r' ↔ a < b 
goal: (AffineMap.lineMap a b) r < (AffineMap.lineMap a b) r' ↔ a < b

full_name: left_lt_lineMap_iff_lt
code: theorem left_lt_lineMap_iff_lt (h : 0 < r) : a < lineMap a b r ↔ a < b 
goal: a < (AffineMap.lineMap a b) r ↔ a < b

full_name: lineMap_lt_left_iff_lt
code: theorem lineMap_lt_left_iff_lt (h : 0 < r) : lineMap a b r < a ↔ b < a 
goal: (AffineMap.lineMap a b) r < a ↔ b < a

full_name: lineMap_lt_right_iff_lt
code: theorem lineMap_lt_right_iff_lt (h : r < 1) : lineMap a b r < b ↔ a < b 
goal: (AffineMap.lineMap a b) r < b ↔ a < b

full_name: right_lt_lineMap_iff_lt
code: theorem right_lt_lineMap_iff_lt (h : r < 1) : b < lineMap a b r ↔ b < a 
goal: b < (AffineMap.lineMap a b) r ↔ b < a

full_name: midpoint_le_midpoint
code: theorem midpoint_le_midpoint (ha : a ≤ a') (hb : b ≤ b') : midpoint k a b ≤ midpoint k a' b' 
goal: midpoint k a b ≤ midpoint k a' b'

full_name: lineMap_le_lineMap_iff_of_lt
code: theorem lineMap_le_lineMap_iff_of_lt (h : r < r') : lineMap a b r ≤ lineMap a b r' ↔ a ≤ b 
goal: (AffineMap.lineMap a b) r ≤ (AffineMap.lineMap a b) r' ↔ a ≤ b

full_name: left_le_lineMap_iff_le
code: theorem left_le_lineMap_iff_le (h : 0 < r) : a ≤ lineMap a b r ↔ a ≤ b 
goal: a ≤ (AffineMap.lineMap a b) r ↔ a ≤ b

full_name: left_le_midpoint
code: @[simp]
theorem left_le_midpoint : a ≤ midpoint k a b ↔ a ≤ b 
goal: a ≤ midpoint k a b ↔ a ≤ b

full_name: lineMap_le_left_iff_le
code: theorem lineMap_le_left_iff_le (h : 0 < r) : lineMap a b r ≤ a ↔ b ≤ a 
goal: (AffineMap.lineMap a b) r ≤ a ↔ b ≤ a

full_name: midpoint_le_left
code: @[simp]
theorem midpoint_le_left : midpoint k a b ≤ a ↔ b ≤ a 
goal: midpoint k a b ≤ a ↔ b ≤ a

full_name: lineMap_le_right_iff_le
code: theorem lineMap_le_right_iff_le (h : r < 1) : lineMap a b r ≤ b ↔ a ≤ b 
goal: (AffineMap.lineMap a b) r ≤ b ↔ a ≤ b

full_name: midpoint_le_right
code: @[simp]
theorem midpoint_le_right : midpoint k a b ≤ b ↔ a ≤ b 
goal: midpoint k a b ≤ b ↔ a ≤ b

full_name: right_le_lineMap_iff_le
code: theorem right_le_lineMap_iff_le (h : r < 1) : b ≤ lineMap a b r ↔ b ≤ a 
goal: b ≤ (AffineMap.lineMap a b) r ↔ b ≤ a

full_name: right_le_midpoint
code: @[simp]
theorem right_le_midpoint : b ≤ midpoint k a b ↔ b ≤ a 
goal: b ≤ midpoint k a b ↔ b ≤ a

full_name: _aux_Mathlib_LinearAlgebra_AffineSpace_Ordered___macroRules_termC_1
code: local notation "c" => lineMap a b r
goal: Lean.Macro

full_name: _aux_Mathlib_LinearAlgebra_AffineSpace_Ordered___unexpand_AffineMap_lineMap_1
code: local notation "c" => lineMap a b r
goal: Lean.PrettyPrinter.Unexpander

full_name: map_le_lineMap_iff_slope_le_slope_left
code: /-- Given `c = lineMap a b r`, `a < c`, the point `(c, f c)` is non-strictly below the
segment `[(a, f a), (b, f b)]` if and only if `slope f a c ≤ slope f a b`. -/
theorem map_le_lineMap_iff_slope_le_slope_left (h : 0 < r * (b - a)) :
    f c ≤ lineMap (f a) (f b) r ↔ slope f a c ≤ slope f a b 
goal: f ((AffineMap.lineMap a b) r) ≤ (AffineMap.lineMap (f a) (f b)) r ↔ slope f a ((AffineMap.lineMap a b) r) ≤ slope f a b

full_name: lineMap_le_map_iff_slope_le_slope_left
code: /-- Given `c = lineMap a b r`, `a < c`, the point `(c, f c)` is non-strictly above the
segment `[(a, f a), (b, f b)]` if and only if `slope f a b ≤ slope f a c`. -/
theorem lineMap_le_map_iff_slope_le_slope_left (h : 0 < r * (b - a)) :
    lineMap (f a) (f b) r ≤ f c ↔ slope f a b ≤ slope f a c 
goal: (AffineMap.lineMap (f a) (f b)) r ≤ f ((AffineMap.lineMap a b) r) ↔ slope f a b ≤ slope f a ((AffineMap.lineMap a b) r)

full_name: map_lt_lineMap_iff_slope_lt_slope_left
code: /-- Given `c = lineMap a b r`, `a < c`, the point `(c, f c)` is strictly below the
segment `[(a, f a), (b, f b)]` if and only if `slope f a c < slope f a b`. -/
theorem map_lt_lineMap_iff_slope_lt_slope_left (h : 0 < r * (b - a)) :
    f c < lineMap (f a) (f b) r ↔ slope f a c < slope f a b 
goal: f ((AffineMap.lineMap a b) r) < (AffineMap.lineMap (f a) (f b)) r ↔ slope f a ((AffineMap.lineMap a b) r) < slope f a b

full_name: lineMap_lt_map_iff_slope_lt_slope_left
code: /-- Given `c = lineMap a b r`, `a < c`, the point `(c, f c)` is strictly above the
segment `[(a, f a), (b, f b)]` if and only if `slope f a b < slope f a c`. -/
theorem lineMap_lt_map_iff_slope_lt_slope_left (h : 0 < r * (b - a)) :
    lineMap (f a) (f b) r < f c ↔ slope f a b < slope f a c 
goal: (AffineMap.lineMap (f a) (f b)) r < f ((AffineMap.lineMap a b) r) ↔ slope f a b < slope f a ((AffineMap.lineMap a b) r)

full_name: map_le_lineMap_iff_slope_le_slope_right
code: /-- Given `c = lineMap a b r`, `c < b`, the point `(c, f c)` is non-strictly below the
segment `[(a, f a), (b, f b)]` if and only if `slope f a b ≤ slope f c b`. -/
theorem map_le_lineMap_iff_slope_le_slope_right (h : 0 < (1 - r) * (b - a)) :
    f c ≤ lineMap (f a) (f b) r ↔ slope f a b ≤ slope f c b 
goal: f ((AffineMap.lineMap a b) r) ≤ (AffineMap.lineMap (f a) (f b)) r ↔ slope f a b ≤ slope f ((AffineMap.lineMap a b) r) b

full_name: lineMap_le_map_iff_slope_le_slope_right
code: /-- Given `c = lineMap a b r`, `c < b`, the point `(c, f c)` is non-strictly above the
segment `[(a, f a), (b, f b)]` if and only if `slope f c b ≤ slope f a b`. -/
theorem lineMap_le_map_iff_slope_le_slope_right (h : 0 < (1 - r) * (b - a)) :
    lineMap (f a) (f b) r ≤ f c ↔ slope f c b ≤ slope f a b 
goal: (AffineMap.lineMap (f a) (f b)) r ≤ f ((AffineMap.lineMap a b) r) ↔ slope f ((AffineMap.lineMap a b) r) b ≤ slope f a b

full_name: map_lt_lineMap_iff_slope_lt_slope_right
code: /-- Given `c = lineMap a b r`, `c < b`, the point `(c, f c)` is strictly below the
segment `[(a, f a), (b, f b)]` if and only if `slope f a b < slope f c b`. -/
theorem map_lt_lineMap_iff_slope_lt_slope_right (h : 0 < (1 - r) * (b - a)) :
    f c < lineMap (f a) (f b) r ↔ slope f a b < slope f c b 
goal: f ((AffineMap.lineMap a b) r) < (AffineMap.lineMap (f a) (f b)) r ↔ slope f a b < slope f ((AffineMap.lineMap a b) r) b

full_name: lineMap_lt_map_iff_slope_lt_slope_right
code: /-- Given `c = lineMap a b r`, `c < b`, the point `(c, f c)` is strictly above the
segment `[(a, f a), (b, f b)]` if and only if `slope f c b < slope f a b`. -/
theorem lineMap_lt_map_iff_slope_lt_slope_right (h : 0 < (1 - r) * (b - a)) :
    lineMap (f a) (f b) r < f c ↔ slope f c b < slope f a b 
goal: (AffineMap.lineMap (f a) (f b)) r < f ((AffineMap.lineMap a b) r) ↔ slope f ((AffineMap.lineMap a b) r) b < slope f a b

full_name: map_le_lineMap_iff_slope_le_slope
code: /-- Given `c = lineMap a b r`, `a < c < b`, the point `(c, f c)` is non-strictly below the
segment `[(a, f a), (b, f b)]` if and only if `slope f a c ≤ slope f c b`. -/
theorem map_le_lineMap_iff_slope_le_slope (hab : a < b) (h₀ : 0 < r) (h₁ : r < 1) :
    f c ≤ lineMap (f a) (f b) r ↔ slope f a c ≤ slope f c b 
goal: f ((AffineMap.lineMap a b) r) ≤ (AffineMap.lineMap (f a) (f b)) r ↔
  slope f a ((AffineMap.lineMap a b) r) ≤ slope f ((AffineMap.lineMap a b) r) b

full_name: lineMap_le_map_iff_slope_le_slope
code: /-- Given `c = lineMap a b r`, `a < c < b`, the point `(c, f c)` is non-strictly above the
segment `[(a, f a), (b, f b)]` if and only if `slope f c b ≤ slope f a c`. -/
theorem lineMap_le_map_iff_slope_le_slope (hab : a < b) (h₀ : 0 < r) (h₁ : r < 1) :
    lineMap (f a) (f b) r ≤ f c ↔ slope f c b ≤ slope f a c 
goal: (AffineMap.lineMap (f a) (f b)) r ≤ f ((AffineMap.lineMap a b) r) ↔
  slope f ((AffineMap.lineMap a b) r) b ≤ slope f a ((AffineMap.lineMap a b) r)

full_name: map_lt_lineMap_iff_slope_lt_slope
code: /-- Given `c = lineMap a b r`, `a < c < b`, the point `(c, f c)` is strictly below the
segment `[(a, f a), (b, f b)]` if and only if `slope f a c < slope f c b`. -/
theorem map_lt_lineMap_iff_slope_lt_slope (hab : a < b) (h₀ : 0 < r) (h₁ : r < 1) :
    f c < lineMap (f a) (f b) r ↔ slope f a c < slope f c b 
goal: f ((AffineMap.lineMap a b) r) < (AffineMap.lineMap (f a) (f b)) r ↔
  slope f a ((AffineMap.lineMap a b) r) < slope f ((AffineMap.lineMap a b) r) b

full_name: lineMap_lt_map_iff_slope_lt_slope
code: /-- Given `c = lineMap a b r`, `a < c < b`, the point `(c, f c)` is strictly above the
segment `[(a, f a), (b, f b)]` if and only if `slope f c b < slope f a c`. -/
theorem lineMap_lt_map_iff_slope_lt_slope (hab : a < b) (h₀ : 0 < r) (h₁ : r < 1) :
    lineMap (f a) (f b) r < f c ↔ slope f c b < slope f a c 
goal: (AffineMap.lineMap (f a) (f b)) r < f ((AffineMap.lineMap a b) r) ↔
  slope f ((AffineMap.lineMap a b) r) b < slope f a ((AffineMap.lineMap a b) r)