[Merged by Bors] - refactor(Analysis): golf Mathlib/Analysis/SpecialFunctions/Log/Monotone

Mathlib/Analysis/SpecialFunctions/Log/Monotone.lean

@@ -57,24 +57,16 @@ theorem log_div_self_rpow_antitoneOn {a : ℝ} (ha : 0 < a) :
   intro x hex y _ hxy
   have x_pos : 0 < x := lt_of_lt_of_le (exp_pos (1 / a)) hex
   have y_pos : 0 < y := by linarith
-  nth_rw 1 [← rpow_one y]
-  nth_rw 1 [← rpow_one x]
+  nth_rw 1 [← rpow_one y, ← rpow_one x]
   rw [← div_self (ne_of_lt ha).symm, div_eq_mul_one_div a a, rpow_mul y_pos.le, rpow_mul x_pos.le,
     log_rpow (rpow_pos_of_pos y_pos a), log_rpow (rpow_pos_of_pos x_pos a), mul_div_assoc,
     mul_div_assoc, mul_le_mul_iff_right₀ (one_div_pos.mpr ha)]
-  refine log_div_self_antitoneOn ?_ ?_ ?_
-  · simp only [Set.mem_setOf_eq]
-    convert rpow_le_rpow _ hex (le_of_lt ha) using 1
-    · rw [← exp_mul]
-      simp only [Real.exp_eq_exp]
-      field
-    positivity
-  · simp only [Set.mem_setOf_eq]
-    convert rpow_le_rpow _ (_root_.trans hex hxy) (le_of_lt ha) using 1
-    · rw [← exp_mul]
-      simp only [Real.exp_eq_exp]
-      field
+  have hbound {z : ℝ} (hz : rexp (1 / a) ≤ z) : z ^ a ∈ {b | rexp 1 ≤ b} := by
+    rw [mem_setOf_eq]
+    convert rpow_le_rpow _ hz (le_of_lt ha) using 1
+    · simp only [← exp_mul, Real.exp_eq_exp, field]
     positivity
+  refine log_div_self_antitoneOn (hbound hex) (hbound (hex.trans hxy)) ?_
   gcongr
 
 theorem log_div_sqrt_antitoneOn : AntitoneOn (fun x : ℝ => log x / √x) { x | exp 2 ≤ x } := by