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Metamath Proof Explorer

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 ) ) )