Theorem leadd2

Description: Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999)

	Ref	Expression
Assertion	leadd2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 + 𝐴 ) ≤ ( 𝐶 + 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		leadd1	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 + 𝐶 ) ≤ ( 𝐵 + 𝐶 ) ) )
2		simp1	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐴 ∈ ℝ )
3	2	recnd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐴 ∈ ℂ )
4		simp3	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐶 ∈ ℝ )
5	4	recnd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐶 ∈ ℂ )
6	3 5	addcomd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 + 𝐶 ) = ( 𝐶 + 𝐴 ) )
7		simp2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐵 ∈ ℝ )
8	7	recnd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐵 ∈ ℂ )
9	8 5	addcomd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 + 𝐶 ) = ( 𝐶 + 𝐵 ) )
10	6 9	breq12d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 + 𝐶 ) ≤ ( 𝐵 + 𝐶 ) ↔ ( 𝐶 + 𝐴 ) ≤ ( 𝐶 + 𝐵 ) ) )
11	1 10	bitrd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 + 𝐴 ) ≤ ( 𝐶 + 𝐵 ) ) )