Step |
Hyp |
Ref |
Expression |
1 |
|
neghalfpire |
|
2 |
1
|
rexri |
|
3 |
|
halfpire |
|
4 |
3
|
rexri |
|
5 |
|
pirp |
|
6 |
|
rphalfcl |
|
7 |
5 6
|
ax-mp |
|
8 |
|
rpgt0 |
|
9 |
7 8
|
ax-mp |
|
10 |
|
lt0neg2 |
|
11 |
3 10
|
ax-mp |
|
12 |
9 11
|
mpbi |
|
13 |
|
0re |
|
14 |
1 13 3
|
lttri |
|
15 |
12 9 14
|
mp2an |
|
16 |
1 3 15
|
ltleii |
|
17 |
|
prunioo |
|
18 |
2 4 16 17
|
mp3an |
|
19 |
18
|
eleq2i |
|
20 |
|
elun |
|
21 |
19 20
|
bitr3i |
|
22 |
|
elioore |
|
23 |
22
|
recnd |
|
24 |
22
|
rered |
|
25 |
|
id |
|
26 |
24 25
|
eqeltrd |
|
27 |
|
asinsin |
|
28 |
23 26 27
|
syl2anc |
|
29 |
|
elpri |
|
30 |
|
ax-1cn |
|
31 |
|
asinneg |
|
32 |
30 31
|
ax-mp |
|
33 |
|
asin1 |
|
34 |
33
|
negeqi |
|
35 |
32 34
|
eqtri |
|
36 |
|
fveq2 |
|
37 |
3
|
recni |
|
38 |
|
sinneg |
|
39 |
37 38
|
ax-mp |
|
40 |
|
sinhalfpi |
|
41 |
40
|
negeqi |
|
42 |
39 41
|
eqtri |
|
43 |
36 42
|
eqtrdi |
|
44 |
43
|
fveq2d |
|
45 |
|
id |
|
46 |
35 44 45
|
3eqtr4a |
|
47 |
|
fveq2 |
|
48 |
47 40
|
eqtrdi |
|
49 |
48
|
fveq2d |
|
50 |
|
id |
|
51 |
33 49 50
|
3eqtr4a |
|
52 |
46 51
|
jaoi |
|
53 |
29 52
|
syl |
|
54 |
28 53
|
jaoi |
|
55 |
21 54
|
sylbi |
|