Theorem: C03S01.my_lemma4 (LeanCourse/MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean)

State before:
⊢ ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε
Tactic:
intro x y ε epos ele1 xlt ylt
State after:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε

State before:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε
Tactic:
calc
  |x * y| = |x| * |y| := by apply abs_mul
  _ ≤ |x| * ε := by apply mul_le_mul; linarith; linarith; apply abs_nonneg; apply abs_nonneg;
  _ < 1 * ε := by rw [mul_lt_mul_right epos]; linarith
  _ = ε := by apply one_mul
State after:
no goals

State before:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| = |x| * |y|
Tactic:
apply abs_mul
State after:
no goals

State before:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| * |y| ≤ |x| * ε
Tactic:
apply mul_le_mul
State after:
case h₁
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| ≤ |x|

case h₂
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |y| ≤ ε

case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|

case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x|

State before:
case h₁
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| ≤ |x|

case h₂
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |y| ≤ ε

case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|

case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x|
Tactic:
linarith
State after:
case h₂
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |y| ≤ ε

case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|

case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x|

State before:
case h₂
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |y| ≤ ε

case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|

case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x|
Tactic:
linarith
State after:
case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|

case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x|

State before:
case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|

case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x|
Tactic:
apply abs_nonneg
State after:
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x|

State before:
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x|
Tactic:
apply abs_nonneg
State after:
no goals

State before:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| * ε < 1 * ε
Tactic:
rw [mul_lt_mul_right epos]
State after:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| < 1

State before:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| < 1
Tactic:
linarith
State after:
no goals

State before:
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 1 * ε = ε
Tactic:
apply one_mul
State after:
no goals