Step |
Hyp |
Ref |
Expression |
1 |
|
cosord.1 |
|- ( ph -> A e. ( 0 [,] _pi ) ) |
2 |
|
cosord.2 |
|- ( ph -> B e. ( 0 [,] _pi ) ) |
3 |
|
cosord.3 |
|- ( ph -> A < B ) |
4 |
|
0re |
|- 0 e. RR |
5 |
|
pire |
|- _pi e. RR |
6 |
4 5
|
elicc2i |
|- ( B e. ( 0 [,] _pi ) <-> ( B e. RR /\ 0 <_ B /\ B <_ _pi ) ) |
7 |
2 6
|
sylib |
|- ( ph -> ( B e. RR /\ 0 <_ B /\ B <_ _pi ) ) |
8 |
7
|
simp1d |
|- ( ph -> B e. RR ) |
9 |
8
|
recnd |
|- ( ph -> B e. CC ) |
10 |
4 5
|
elicc2i |
|- ( A e. ( 0 [,] _pi ) <-> ( A e. RR /\ 0 <_ A /\ A <_ _pi ) ) |
11 |
1 10
|
sylib |
|- ( ph -> ( A e. RR /\ 0 <_ A /\ A <_ _pi ) ) |
12 |
11
|
simp1d |
|- ( ph -> A e. RR ) |
13 |
12
|
recnd |
|- ( ph -> A e. CC ) |
14 |
|
subcos |
|- ( ( B e. CC /\ A e. CC ) -> ( ( cos ` A ) - ( cos ` B ) ) = ( 2 x. ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) ) |
15 |
9 13 14
|
syl2anc |
|- ( ph -> ( ( cos ` A ) - ( cos ` B ) ) = ( 2 x. ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) ) |
16 |
|
2rp |
|- 2 e. RR+ |
17 |
8 12
|
readdcld |
|- ( ph -> ( B + A ) e. RR ) |
18 |
17
|
rehalfcld |
|- ( ph -> ( ( B + A ) / 2 ) e. RR ) |
19 |
18
|
resincld |
|- ( ph -> ( sin ` ( ( B + A ) / 2 ) ) e. RR ) |
20 |
4
|
a1i |
|- ( ph -> 0 e. RR ) |
21 |
11
|
simp2d |
|- ( ph -> 0 <_ A ) |
22 |
20 12 8 21 3
|
lelttrd |
|- ( ph -> 0 < B ) |
23 |
8 12 22 21
|
addgtge0d |
|- ( ph -> 0 < ( B + A ) ) |
24 |
|
2re |
|- 2 e. RR |
25 |
|
2pos |
|- 0 < 2 |
26 |
|
divgt0 |
|- ( ( ( ( B + A ) e. RR /\ 0 < ( B + A ) ) /\ ( 2 e. RR /\ 0 < 2 ) ) -> 0 < ( ( B + A ) / 2 ) ) |
27 |
24 25 26
|
mpanr12 |
|- ( ( ( B + A ) e. RR /\ 0 < ( B + A ) ) -> 0 < ( ( B + A ) / 2 ) ) |
28 |
17 23 27
|
syl2anc |
|- ( ph -> 0 < ( ( B + A ) / 2 ) ) |
29 |
5
|
a1i |
|- ( ph -> _pi e. RR ) |
30 |
12 8 8 3
|
ltadd2dd |
|- ( ph -> ( B + A ) < ( B + B ) ) |
31 |
9
|
2timesd |
|- ( ph -> ( 2 x. B ) = ( B + B ) ) |
32 |
30 31
|
breqtrrd |
|- ( ph -> ( B + A ) < ( 2 x. B ) ) |
33 |
24
|
a1i |
|- ( ph -> 2 e. RR ) |
34 |
25
|
a1i |
|- ( ph -> 0 < 2 ) |
35 |
|
ltdivmul |
|- ( ( ( B + A ) e. RR /\ B e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( ( B + A ) / 2 ) < B <-> ( B + A ) < ( 2 x. B ) ) ) |
36 |
17 8 33 34 35
|
syl112anc |
|- ( ph -> ( ( ( B + A ) / 2 ) < B <-> ( B + A ) < ( 2 x. B ) ) ) |
37 |
32 36
|
mpbird |
|- ( ph -> ( ( B + A ) / 2 ) < B ) |
38 |
7
|
simp3d |
|- ( ph -> B <_ _pi ) |
39 |
18 8 29 37 38
|
ltletrd |
|- ( ph -> ( ( B + A ) / 2 ) < _pi ) |
40 |
|
0xr |
|- 0 e. RR* |
41 |
5
|
rexri |
|- _pi e. RR* |
42 |
|
elioo2 |
|- ( ( 0 e. RR* /\ _pi e. RR* ) -> ( ( ( B + A ) / 2 ) e. ( 0 (,) _pi ) <-> ( ( ( B + A ) / 2 ) e. RR /\ 0 < ( ( B + A ) / 2 ) /\ ( ( B + A ) / 2 ) < _pi ) ) ) |
43 |
40 41 42
|
mp2an |
|- ( ( ( B + A ) / 2 ) e. ( 0 (,) _pi ) <-> ( ( ( B + A ) / 2 ) e. RR /\ 0 < ( ( B + A ) / 2 ) /\ ( ( B + A ) / 2 ) < _pi ) ) |
44 |
18 28 39 43
|
syl3anbrc |
|- ( ph -> ( ( B + A ) / 2 ) e. ( 0 (,) _pi ) ) |
45 |
|
sinq12gt0 |
|- ( ( ( B + A ) / 2 ) e. ( 0 (,) _pi ) -> 0 < ( sin ` ( ( B + A ) / 2 ) ) ) |
46 |
44 45
|
syl |
|- ( ph -> 0 < ( sin ` ( ( B + A ) / 2 ) ) ) |
47 |
19 46
|
elrpd |
|- ( ph -> ( sin ` ( ( B + A ) / 2 ) ) e. RR+ ) |
48 |
8 12
|
resubcld |
|- ( ph -> ( B - A ) e. RR ) |
49 |
48
|
rehalfcld |
|- ( ph -> ( ( B - A ) / 2 ) e. RR ) |
50 |
49
|
resincld |
|- ( ph -> ( sin ` ( ( B - A ) / 2 ) ) e. RR ) |
51 |
12 8
|
posdifd |
|- ( ph -> ( A < B <-> 0 < ( B - A ) ) ) |
52 |
3 51
|
mpbid |
|- ( ph -> 0 < ( B - A ) ) |
53 |
|
divgt0 |
|- ( ( ( ( B - A ) e. RR /\ 0 < ( B - A ) ) /\ ( 2 e. RR /\ 0 < 2 ) ) -> 0 < ( ( B - A ) / 2 ) ) |
54 |
24 25 53
|
mpanr12 |
|- ( ( ( B - A ) e. RR /\ 0 < ( B - A ) ) -> 0 < ( ( B - A ) / 2 ) ) |
55 |
48 52 54
|
syl2anc |
|- ( ph -> 0 < ( ( B - A ) / 2 ) ) |
56 |
|
rehalfcl |
|- ( _pi e. RR -> ( _pi / 2 ) e. RR ) |
57 |
5 56
|
mp1i |
|- ( ph -> ( _pi / 2 ) e. RR ) |
58 |
8 12
|
subge02d |
|- ( ph -> ( 0 <_ A <-> ( B - A ) <_ B ) ) |
59 |
21 58
|
mpbid |
|- ( ph -> ( B - A ) <_ B ) |
60 |
48 8 29 59 38
|
letrd |
|- ( ph -> ( B - A ) <_ _pi ) |
61 |
|
lediv1 |
|- ( ( ( B - A ) e. RR /\ _pi e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( B - A ) <_ _pi <-> ( ( B - A ) / 2 ) <_ ( _pi / 2 ) ) ) |
62 |
48 29 33 34 61
|
syl112anc |
|- ( ph -> ( ( B - A ) <_ _pi <-> ( ( B - A ) / 2 ) <_ ( _pi / 2 ) ) ) |
63 |
60 62
|
mpbid |
|- ( ph -> ( ( B - A ) / 2 ) <_ ( _pi / 2 ) ) |
64 |
|
pirp |
|- _pi e. RR+ |
65 |
|
rphalflt |
|- ( _pi e. RR+ -> ( _pi / 2 ) < _pi ) |
66 |
64 65
|
mp1i |
|- ( ph -> ( _pi / 2 ) < _pi ) |
67 |
49 57 29 63 66
|
lelttrd |
|- ( ph -> ( ( B - A ) / 2 ) < _pi ) |
68 |
|
elioo2 |
|- ( ( 0 e. RR* /\ _pi e. RR* ) -> ( ( ( B - A ) / 2 ) e. ( 0 (,) _pi ) <-> ( ( ( B - A ) / 2 ) e. RR /\ 0 < ( ( B - A ) / 2 ) /\ ( ( B - A ) / 2 ) < _pi ) ) ) |
69 |
40 41 68
|
mp2an |
|- ( ( ( B - A ) / 2 ) e. ( 0 (,) _pi ) <-> ( ( ( B - A ) / 2 ) e. RR /\ 0 < ( ( B - A ) / 2 ) /\ ( ( B - A ) / 2 ) < _pi ) ) |
70 |
49 55 67 69
|
syl3anbrc |
|- ( ph -> ( ( B - A ) / 2 ) e. ( 0 (,) _pi ) ) |
71 |
|
sinq12gt0 |
|- ( ( ( B - A ) / 2 ) e. ( 0 (,) _pi ) -> 0 < ( sin ` ( ( B - A ) / 2 ) ) ) |
72 |
70 71
|
syl |
|- ( ph -> 0 < ( sin ` ( ( B - A ) / 2 ) ) ) |
73 |
50 72
|
elrpd |
|- ( ph -> ( sin ` ( ( B - A ) / 2 ) ) e. RR+ ) |
74 |
47 73
|
rpmulcld |
|- ( ph -> ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) e. RR+ ) |
75 |
|
rpmulcl |
|- ( ( 2 e. RR+ /\ ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) e. RR+ ) -> ( 2 x. ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) e. RR+ ) |
76 |
16 74 75
|
sylancr |
|- ( ph -> ( 2 x. ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) e. RR+ ) |
77 |
15 76
|
eqeltrd |
|- ( ph -> ( ( cos ` A ) - ( cos ` B ) ) e. RR+ ) |
78 |
8
|
recoscld |
|- ( ph -> ( cos ` B ) e. RR ) |
79 |
12
|
recoscld |
|- ( ph -> ( cos ` A ) e. RR ) |
80 |
|
difrp |
|- ( ( ( cos ` B ) e. RR /\ ( cos ` A ) e. RR ) -> ( ( cos ` B ) < ( cos ` A ) <-> ( ( cos ` A ) - ( cos ` B ) ) e. RR+ ) ) |
81 |
78 79 80
|
syl2anc |
|- ( ph -> ( ( cos ` B ) < ( cos ` A ) <-> ( ( cos ` A ) - ( cos ` B ) ) e. RR+ ) ) |
82 |
77 81
|
mpbird |
|- ( ph -> ( cos ` B ) < ( cos ` A ) ) |