This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem leaddsub2

Description: 'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005)

Ref Expression
Assertion leaddsub2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 + 𝐵 ) ≤ 𝐶 ↔ 𝐵 ≤ ( 𝐶 − 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 recn ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
2 recn ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
3 addcom ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
5 4 3adant3 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
6 5 breq1d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 + 𝐵 ) ≤ 𝐶 ↔ ( 𝐵 + 𝐴 ) ≤ 𝐶 ) )
7 leaddsub ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐵 + 𝐴 ) ≤ 𝐶 ↔ 𝐵 ≤ ( 𝐶 − 𝐴 ) ) )
8 7 3com12 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐵 + 𝐴 ) ≤ 𝐶 ↔ 𝐵 ≤ ( 𝐶 − 𝐴 ) ) )
9 6 8 bitrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 + 𝐵 ) ≤ 𝐶 ↔ 𝐵 ≤ ( 𝐶 − 𝐴 ) ) )