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feat(Data/Real/EReal): add mul_lt_mul_of_pos_left theorem #31796

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feat(Data/Real/EReal): add mul_lt_mul_of_pos_left theorem #31796

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23 changes: 23 additions & 0 deletions Mathlib/Data/EReal/Operations.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -822,6 +822,29 @@ lemma nsmul_eq_mul (n : ℕ) (x : EReal) : n • x = n * x := by
| succ n ih =>
rw [succ_nsmul, ih, Nat.cast_succ]
convert (EReal.right_distrib_of_nonneg _ _).symm <;> simp
/-! ### Multiplication and order -/
theorem mul_lt_mul_of_pos_left {a : ℝ} {b c : EReal} (ha_pos : 0 < a) (hbc : b < c) :
a.toEReal * b < a.toEReal * c :=by
apply EReal.sub_pos.mp
rw [sub_eq_add_neg]
rw [← mul_neg]
rw [← EReal.left_distrib_of_nonneg_of_ne_top]
·have h1 : 0 < ↑a * ( c+ (-b)):=by
apply EReal.mul_pos_iff.mpr
left
constructor
·apply EReal.coe_pos.mpr
apply ha_pos
·apply sub_pos.mpr
apply hbc
apply h1
·have h2 : 0 ≤ Real.toEReal a :=by
apply EReal.coe_nonneg.mpr
apply le_of_lt
apply ha_pos
apply h2
·apply EReal.coe_ne_top


end EReal

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