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 ) ) )