Step |
Hyp |
Ref |
Expression |
1 |
|
0z |
|- 0 e. ZZ |
2 |
|
nnre |
|- ( B e. NN -> B e. RR ) |
3 |
2
|
3ad2ant2 |
|- ( ( A e. RR /\ B e. NN /\ B < A ) -> B e. RR ) |
4 |
|
simp1 |
|- ( ( A e. RR /\ B e. NN /\ B < A ) -> A e. RR ) |
5 |
|
nngt0 |
|- ( B e. NN -> 0 < B ) |
6 |
5
|
3ad2ant2 |
|- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < B ) |
7 |
5
|
adantl |
|- ( ( A e. RR /\ B e. NN ) -> 0 < B ) |
8 |
|
0re |
|- 0 e. RR |
9 |
|
lttr |
|- ( ( 0 e. RR /\ B e. RR /\ A e. RR ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) ) |
10 |
8 9
|
mp3an1 |
|- ( ( B e. RR /\ A e. RR ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) ) |
11 |
2 10
|
sylan |
|- ( ( B e. NN /\ A e. RR ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) ) |
12 |
11
|
ancoms |
|- ( ( A e. RR /\ B e. NN ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) ) |
13 |
7 12
|
mpand |
|- ( ( A e. RR /\ B e. NN ) -> ( B < A -> 0 < A ) ) |
14 |
13
|
3impia |
|- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < A ) |
15 |
3 4 6 14
|
divgt0d |
|- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < ( B / A ) ) |
16 |
|
simp3 |
|- ( ( A e. RR /\ B e. NN /\ B < A ) -> B < A ) |
17 |
|
1re |
|- 1 e. RR |
18 |
|
ltdivmul2 |
|- ( ( B e. RR /\ 1 e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) ) |
19 |
17 18
|
mp3an2 |
|- ( ( B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) ) |
20 |
3 4 14 19
|
syl12anc |
|- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) ) |
21 |
|
recn |
|- ( A e. RR -> A e. CC ) |
22 |
21
|
mulid2d |
|- ( A e. RR -> ( 1 x. A ) = A ) |
23 |
22
|
breq2d |
|- ( A e. RR -> ( B < ( 1 x. A ) <-> B < A ) ) |
24 |
23
|
3ad2ant1 |
|- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( B < ( 1 x. A ) <-> B < A ) ) |
25 |
20 24
|
bitrd |
|- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( ( B / A ) < 1 <-> B < A ) ) |
26 |
16 25
|
mpbird |
|- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( B / A ) < 1 ) |
27 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
28 |
26 27
|
breqtrrdi |
|- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( B / A ) < ( 0 + 1 ) ) |
29 |
|
btwnnz |
|- ( ( 0 e. ZZ /\ 0 < ( B / A ) /\ ( B / A ) < ( 0 + 1 ) ) -> -. ( B / A ) e. ZZ ) |
30 |
1 15 28 29
|
mp3an2i |
|- ( ( A e. RR /\ B e. NN /\ B < A ) -> -. ( B / A ) e. ZZ ) |