Theorem le2halvesd

Description: A sum is less than the whole if each term is less than half. (Contributed by Thierry Arnoux, 29-Nov-2017)

	Ref	Expression
Hypotheses	le2halvesd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	le2halvesd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	le2halvesd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	le2halvesd.4	⊢ ( 𝜑 → 𝐴 ≤ ( 𝐶 / 2 ) )
	le2halvesd.5	⊢ ( 𝜑 → 𝐵 ≤ ( 𝐶 / 2 ) )
Assertion	le2halvesd	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ≤ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		le2halvesd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		le2halvesd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		le2halvesd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		le2halvesd.4	⊢ ( 𝜑 → 𝐴 ≤ ( 𝐶 / 2 ) )
5		le2halvesd.5	⊢ ( 𝜑 → 𝐵 ≤ ( 𝐶 / 2 ) )
6	3	rehalfcld	⊢ ( 𝜑 → ( 𝐶 / 2 ) ∈ ℝ )
7	1 2 6 6 4 5	le2addd	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ≤ ( ( 𝐶 / 2 ) + ( 𝐶 / 2 ) ) )
8	3	recnd	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
9	8	2halvesd	⊢ ( 𝜑 → ( ( 𝐶 / 2 ) + ( 𝐶 / 2 ) ) = 𝐶 )
10	7 9	breqtrd	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ≤ 𝐶 )