Step |
Hyp |
Ref |
Expression |
1 |
|
pire |
|
2 |
|
3re |
|
3 |
|
halfpire |
|
4 |
2 3
|
remulcli |
|
5 |
|
rexr |
|
6 |
|
rexr |
|
7 |
|
elioo2 |
|
8 |
5 6 7
|
syl2an |
|
9 |
1 4 8
|
mp2an |
|
10 |
|
pidiv2halves |
|
11 |
10
|
breq1i |
|
12 |
|
ltaddsub |
|
13 |
3 3 12
|
mp3an12 |
|
14 |
11 13
|
bitr3id |
|
15 |
|
ltsubadd |
|
16 |
3 1 15
|
mp3an23 |
|
17 |
|
df-3 |
|
18 |
17
|
oveq1i |
|
19 |
|
2cn |
|
20 |
|
ax-1cn |
|
21 |
3
|
recni |
|
22 |
19 20 21
|
adddiri |
|
23 |
1
|
recni |
|
24 |
|
2ne0 |
|
25 |
23 19 24
|
divcan2i |
|
26 |
21
|
mulid2i |
|
27 |
25 26
|
oveq12i |
|
28 |
18 22 27
|
3eqtrri |
|
29 |
28
|
breq2i |
|
30 |
16 29
|
bitr2di |
|
31 |
14 30
|
anbi12d |
|
32 |
|
resubcl |
|
33 |
3 32
|
mpan2 |
|
34 |
|
sincosq2sgn |
|
35 |
|
rexr |
|
36 |
|
elioo2 |
|
37 |
35 5 36
|
syl2an |
|
38 |
3 1 37
|
mp2an |
|
39 |
|
ancom |
|
40 |
34 38 39
|
3imtr3i |
|
41 |
33 40
|
syl3an1 |
|
42 |
41
|
3expib |
|
43 |
31 42
|
sylbid |
|
44 |
33
|
resincld |
|
45 |
44
|
lt0neg2d |
|
46 |
45
|
anbi2d |
|
47 |
43 46
|
sylibd |
|
48 |
|
recn |
|
49 |
|
pncan3 |
|
50 |
21 48 49
|
sylancr |
|
51 |
50
|
fveq2d |
|
52 |
33
|
recnd |
|
53 |
|
sinhalfpip |
|
54 |
52 53
|
syl |
|
55 |
51 54
|
eqtr3d |
|
56 |
55
|
breq1d |
|
57 |
50
|
fveq2d |
|
58 |
|
coshalfpip |
|
59 |
52 58
|
syl |
|
60 |
57 59
|
eqtr3d |
|
61 |
60
|
breq1d |
|
62 |
56 61
|
anbi12d |
|
63 |
47 62
|
sylibrd |
|
64 |
63
|
3impib |
|
65 |
9 64
|
sylbi |
|