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