Theorem lt2halvesd

Description: A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	rehalfcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	lt2halvesd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	lt2halvesd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	lt2halvesd.4	⊢ ( 𝜑 → 𝐴 < ( 𝐶 / 2 ) )
	lt2halvesd.5	⊢ ( 𝜑 → 𝐵 < ( 𝐶 / 2 ) )
Assertion	lt2halvesd	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) < 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		rehalfcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		lt2halvesd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		lt2halvesd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		lt2halvesd.4	⊢ ( 𝜑 → 𝐴 < ( 𝐶 / 2 ) )
5		lt2halvesd.5	⊢ ( 𝜑 → 𝐵 < ( 𝐶 / 2 ) )
6		lt2halves	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < ( 𝐶 / 2 ) ∧ 𝐵 < ( 𝐶 / 2 ) ) → ( 𝐴 + 𝐵 ) < 𝐶 ) )
7	1 2 3 6	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 < ( 𝐶 / 2 ) ∧ 𝐵 < ( 𝐶 / 2 ) ) → ( 𝐴 + 𝐵 ) < 𝐶 ) )
8	4 5 7	mp2and	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) < 𝐶 )