Theorem leltadd

Description: Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008)

	Ref	Expression
Assertion	leltadd	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ) ) → ( ( 𝐴 ≤ 𝐶 ∧ 𝐵 < 𝐷 ) → ( 𝐴 + 𝐵 ) < ( 𝐶 + 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltleadd	⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) ∧ ( 𝐷 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ) → ( ( 𝐵 < 𝐷 ∧ 𝐴 ≤ 𝐶 ) → ( 𝐵 + 𝐴 ) < ( 𝐷 + 𝐶 ) ) )
2	1	ancomsd	⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) ∧ ( 𝐷 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ) → ( ( 𝐴 ≤ 𝐶 ∧ 𝐵 < 𝐷 ) → ( 𝐵 + 𝐴 ) < ( 𝐷 + 𝐶 ) ) )
3	2	ancom2s	⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ) ) → ( ( 𝐴 ≤ 𝐶 ∧ 𝐵 < 𝐷 ) → ( 𝐵 + 𝐴 ) < ( 𝐷 + 𝐶 ) ) )
4	3	ancom1s	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ) ) → ( ( 𝐴 ≤ 𝐶 ∧ 𝐵 < 𝐷 ) → ( 𝐵 + 𝐴 ) < ( 𝐷 + 𝐶 ) ) )
5		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
6		recn	⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
7		addcom	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
8	5 6 7	syl2an	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
9		recn	⊢ ( 𝐶 ∈ ℝ → 𝐶 ∈ ℂ )
10		recn	⊢ ( 𝐷 ∈ ℝ → 𝐷 ∈ ℂ )
11		addcom	⊢ ( ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) → ( 𝐶 + 𝐷 ) = ( 𝐷 + 𝐶 ) )
12	9 10 11	syl2an	⊢ ( ( 𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ) → ( 𝐶 + 𝐷 ) = ( 𝐷 + 𝐶 ) )
13	8 12	breqan12d	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ) ) → ( ( 𝐴 + 𝐵 ) < ( 𝐶 + 𝐷 ) ↔ ( 𝐵 + 𝐴 ) < ( 𝐷 + 𝐶 ) ) )
14	4 13	sylibrd	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ) ) → ( ( 𝐴 ≤ 𝐶 ∧ 𝐵 < 𝐷 ) → ( 𝐴 + 𝐵 ) < ( 𝐶 + 𝐷 ) ) )