Metamath Proof Explorer


Theorem cosordlem

Description: Lemma for cosord . (Contributed by Mario Carneiro, 10-May-2014)

Ref Expression
Hypotheses cosord.1 φ A 0 π
cosord.2 φ B 0 π
cosord.3 φ A < B
Assertion cosordlem φ cos B < cos A

Proof

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