Step |
Hyp |
Ref |
Expression |
1 |
|
dp2lt.a |
|- A e. NN0 |
2 |
|
dp2lt.b |
|- B e. RR+ |
3 |
|
dp2ltc.c |
|- C e. NN0 |
4 |
|
dp2ltc.d |
|- D e. RR+ |
5 |
|
dp2ltc.s |
|- B < ; 1 0 |
6 |
|
dp2ltc.l |
|- A < C |
7 |
|
rpssre |
|- RR+ C_ RR |
8 |
7 2
|
sselii |
|- B e. RR |
9 |
|
10re |
|- ; 1 0 e. RR |
10 |
|
10pos |
|- 0 < ; 1 0 |
11 |
|
elrp |
|- ( ; 1 0 e. RR+ <-> ( ; 1 0 e. RR /\ 0 < ; 1 0 ) ) |
12 |
9 10 11
|
mpbir2an |
|- ; 1 0 e. RR+ |
13 |
|
divlt1lt |
|- ( ( B e. RR /\ ; 1 0 e. RR+ ) -> ( ( B / ; 1 0 ) < 1 <-> B < ; 1 0 ) ) |
14 |
8 12 13
|
mp2an |
|- ( ( B / ; 1 0 ) < 1 <-> B < ; 1 0 ) |
15 |
5 14
|
mpbir |
|- ( B / ; 1 0 ) < 1 |
16 |
9 10
|
gt0ne0ii |
|- ; 1 0 =/= 0 |
17 |
8 9 16
|
redivcli |
|- ( B / ; 1 0 ) e. RR |
18 |
|
1re |
|- 1 e. RR |
19 |
1
|
nn0rei |
|- A e. RR |
20 |
|
ltadd2 |
|- ( ( ( B / ; 1 0 ) e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( B / ; 1 0 ) < 1 <-> ( A + ( B / ; 1 0 ) ) < ( A + 1 ) ) ) |
21 |
17 18 19 20
|
mp3an |
|- ( ( B / ; 1 0 ) < 1 <-> ( A + ( B / ; 1 0 ) ) < ( A + 1 ) ) |
22 |
15 21
|
mpbi |
|- ( A + ( B / ; 1 0 ) ) < ( A + 1 ) |
23 |
1
|
nn0zi |
|- A e. ZZ |
24 |
3
|
nn0zi |
|- C e. ZZ |
25 |
|
zltp1le |
|- ( ( A e. ZZ /\ C e. ZZ ) -> ( A < C <-> ( A + 1 ) <_ C ) ) |
26 |
23 24 25
|
mp2an |
|- ( A < C <-> ( A + 1 ) <_ C ) |
27 |
6 26
|
mpbi |
|- ( A + 1 ) <_ C |
28 |
19 17
|
readdcli |
|- ( A + ( B / ; 1 0 ) ) e. RR |
29 |
19 18
|
readdcli |
|- ( A + 1 ) e. RR |
30 |
3
|
nn0rei |
|- C e. RR |
31 |
28 29 30
|
ltletri |
|- ( ( ( A + ( B / ; 1 0 ) ) < ( A + 1 ) /\ ( A + 1 ) <_ C ) -> ( A + ( B / ; 1 0 ) ) < C ) |
32 |
22 27 31
|
mp2an |
|- ( A + ( B / ; 1 0 ) ) < C |
33 |
4 12
|
pm3.2i |
|- ( D e. RR+ /\ ; 1 0 e. RR+ ) |
34 |
|
rpdivcl |
|- ( ( D e. RR+ /\ ; 1 0 e. RR+ ) -> ( D / ; 1 0 ) e. RR+ ) |
35 |
33 34
|
ax-mp |
|- ( D / ; 1 0 ) e. RR+ |
36 |
|
ltaddrp |
|- ( ( C e. RR /\ ( D / ; 1 0 ) e. RR+ ) -> C < ( C + ( D / ; 1 0 ) ) ) |
37 |
30 35 36
|
mp2an |
|- C < ( C + ( D / ; 1 0 ) ) |
38 |
7 4
|
sselii |
|- D e. RR |
39 |
38 9 16
|
redivcli |
|- ( D / ; 1 0 ) e. RR |
40 |
30 39
|
readdcli |
|- ( C + ( D / ; 1 0 ) ) e. RR |
41 |
28 30 40
|
lttri |
|- ( ( ( A + ( B / ; 1 0 ) ) < C /\ C < ( C + ( D / ; 1 0 ) ) ) -> ( A + ( B / ; 1 0 ) ) < ( C + ( D / ; 1 0 ) ) ) |
42 |
32 37 41
|
mp2an |
|- ( A + ( B / ; 1 0 ) ) < ( C + ( D / ; 1 0 ) ) |
43 |
|
df-dp2 |
|- _ A B = ( A + ( B / ; 1 0 ) ) |
44 |
|
df-dp2 |
|- _ C D = ( C + ( D / ; 1 0 ) ) |
45 |
42 43 44
|
3brtr4i |
|- _ A B < _ C D |