Metamath Proof Explorer


Theorem xrge0addgt0

Description: The sum of nonnegative and positive numbers is positive. See addgtge0 . (Contributed by Thierry Arnoux, 6-Jul-2017)

Ref Expression
Assertion xrge0addgt0 A 0 +∞ B 0 +∞ 0 < A 0 < A + 𝑒 B

Proof

Step Hyp Ref Expression
1 0xr 0 *
2 xaddid1 0 * 0 + 𝑒 0 = 0
3 1 2 ax-mp 0 + 𝑒 0 = 0
4 simplr A 0 +∞ B 0 +∞ 0 < A 0 < B 0 < A
5 simpr A 0 +∞ B 0 +∞ 0 < A 0 < B 0 < B
6 1 a1i A 0 +∞ B 0 +∞ 0 < A 0 < B 0 *
7 iccssxr 0 +∞ *
8 simplll A 0 +∞ B 0 +∞ 0 < A 0 < B A 0 +∞
9 7 8 sselid A 0 +∞ B 0 +∞ 0 < A 0 < B A *
10 simpllr A 0 +∞ B 0 +∞ 0 < A 0 < B B 0 +∞
11 7 10 sselid A 0 +∞ B 0 +∞ 0 < A 0 < B B *
12 xlt2add 0 * 0 * A * B * 0 < A 0 < B 0 + 𝑒 0 < A + 𝑒 B
13 6 6 9 11 12 syl22anc A 0 +∞ B 0 +∞ 0 < A 0 < B 0 < A 0 < B 0 + 𝑒 0 < A + 𝑒 B
14 4 5 13 mp2and A 0 +∞ B 0 +∞ 0 < A 0 < B 0 + 𝑒 0 < A + 𝑒 B
15 3 14 eqbrtrrid A 0 +∞ B 0 +∞ 0 < A 0 < B 0 < A + 𝑒 B
16 simplr A 0 +∞ B 0 +∞ 0 < A 0 = B 0 < A
17 oveq2 0 = B A + 𝑒 0 = A + 𝑒 B
18 17 adantl A 0 +∞ B 0 +∞ 0 < A 0 = B A + 𝑒 0 = A + 𝑒 B
19 18 breq2d A 0 +∞ B 0 +∞ 0 < A 0 = B 0 < A + 𝑒 0 0 < A + 𝑒 B
20 simplll A 0 +∞ B 0 +∞ 0 < A 0 = B A 0 +∞
21 7 20 sselid A 0 +∞ B 0 +∞ 0 < A 0 = B A *
22 xaddid1 A * A + 𝑒 0 = A
23 21 22 syl A 0 +∞ B 0 +∞ 0 < A 0 = B A + 𝑒 0 = A
24 23 breq2d A 0 +∞ B 0 +∞ 0 < A 0 = B 0 < A + 𝑒 0 0 < A
25 19 24 bitr3d A 0 +∞ B 0 +∞ 0 < A 0 = B 0 < A + 𝑒 B 0 < A
26 16 25 mpbird A 0 +∞ B 0 +∞ 0 < A 0 = B 0 < A + 𝑒 B
27 1 a1i A 0 +∞ B 0 +∞ 0 < A 0 *
28 simplr A 0 +∞ B 0 +∞ 0 < A B 0 +∞
29 7 28 sselid A 0 +∞ B 0 +∞ 0 < A B *
30 pnfxr +∞ *
31 30 a1i A 0 +∞ B 0 +∞ 0 < A +∞ *
32 iccgelb 0 * +∞ * B 0 +∞ 0 B
33 27 31 28 32 syl3anc A 0 +∞ B 0 +∞ 0 < A 0 B
34 xrleloe 0 * B * 0 B 0 < B 0 = B
35 34 biimpa 0 * B * 0 B 0 < B 0 = B
36 27 29 33 35 syl21anc A 0 +∞ B 0 +∞ 0 < A 0 < B 0 = B
37 15 26 36 mpjaodan A 0 +∞ B 0 +∞ 0 < A 0 < A + 𝑒 B