Metamath Proof Explorer


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
le2halvesd.2 φ B
le2halvesd.3 φ C
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
2 le2halvesd.2 φ B
3 le2halvesd.3 φ C
4 le2halvesd.4 φ A C 2
5 le2halvesd.5 φ B C 2
6 3 rehalfcld φ C 2
7 1 2 6 6 4 5 le2addd φ A + B C 2 + C 2
8 3 recnd φ C
9 8 2halvesd φ C 2 + C 2 = C
10 7 9 breqtrd φ A + B C