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Metamath Proof Explorer

Theorem efle

Description: The exponential function on the reals is nondecreasing. (Contributed by Mario Carneiro, 11-Mar-2014)

Ref Expression
Assertion efle ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( exp ‘ 𝐴 ) ≤ ( exp ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 eflt ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 𝐵 < 𝐴 ↔ ( exp ‘ 𝐵 ) < ( exp ‘ 𝐴 ) ) )
2 1 ancoms ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 < 𝐴 ↔ ( exp ‘ 𝐵 ) < ( exp ‘ 𝐴 ) ) )
3 2 notbid ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ¬ 𝐵 < 𝐴 ↔ ¬ ( exp ‘ 𝐵 ) < ( exp ‘ 𝐴 ) ) )
4 lenlt ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
5 reefcl ⊢ ( 𝐴 ∈ ℝ → ( exp ‘ 𝐴 ) ∈ ℝ )
6 reefcl ⊢ ( 𝐵 ∈ ℝ → ( exp ‘ 𝐵 ) ∈ ℝ )
7 lenlt ⊢ ( ( ( exp ‘ 𝐴 ) ∈ ℝ ∧ ( exp ‘ 𝐵 ) ∈ ℝ ) → ( ( exp ‘ 𝐴 ) ≤ ( exp ‘ 𝐵 ) ↔ ¬ ( exp ‘ 𝐵 ) < ( exp ‘ 𝐴 ) ) )
8 5 6 7 syl2an ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( exp ‘ 𝐴 ) ≤ ( exp ‘ 𝐵 ) ↔ ¬ ( exp ‘ 𝐵 ) < ( exp ‘ 𝐴 ) ) )
9 3 4 8 3bitr4d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( exp ‘ 𝐴 ) ≤ ( exp ‘ 𝐵 ) ) )