Theorem cos12dec 11802

Description: Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.)

Assertion

Ref	Expression
cos12dec	|- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` B ) < ( cos ` A ) )

Proof of Theorem cos12dec

Step	Hyp	Ref	Expression
1		1re 7981	. . . . . . . . . . 11 |- 1 e. RR
2		2re 9014	. . . . . . . . . . 11 |- 2 e. RR
3		iccssre 9980	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ 2 e. RR ) -> ( 1 [,] 2 ) C_ RR )
4	1, 2, 3	mp2an 426	. . . . . . . . . 10 |- ( 1 [,] 2 ) C_ RR
5	4	sseli 3166	. . . . . . . . 9 |- ( B e. ( 1 [,] 2 ) -> B e. RR )
6	5	3ad2ant2 1021	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> B e. RR )
7	6	recnd 8011	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> B e. CC )
8	4	sseli 3166	. . . . . . . . . . 11 |- ( A e. ( 1 [,] 2 ) -> A e. RR )
9	8	3ad2ant1 1020	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> A e. RR )
10	6, 9	resubcld 8363	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) e. RR )
11	10	recnd 8011	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) e. CC )
12	11	halfcld 9188	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) e. CC )
13	7, 12	subcld 8293	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) e. CC )
14	13	coscld 11746	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( B - ( ( B - A ) / 2 ) ) ) e. CC )
15	12	coscld 11746	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - A ) / 2 ) ) e. CC )
16	14, 15	mulcld 8003	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( cos ` ( B - ( ( B - A ) / 2 ) ) ) x. ( cos ` ( ( B - A ) / 2 ) ) ) e. CC )
17	13	sincld 11745	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( B - ( ( B - A ) / 2 ) ) ) e. CC )
18	12	sincld 11745	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( ( B - A ) / 2 ) ) e. CC )
19	17, 18	mulcld 8003	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) e. CC )
20	16, 19	negsubd 8299	. . 3 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( ( cos ` ( B - ( ( B - A ) / 2 ) ) ) x. ( cos ` ( ( B - A ) / 2 ) ) ) + -u ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) = ( ( ( cos ` ( B - ( ( B - A ) / 2 ) ) ) x. ( cos ` ( ( B - A ) / 2 ) ) ) - ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) )
21	10	rehalfcld 9190	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) e. RR )
22	6, 21	resubcld 8363	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) e. RR )
23	22	resincld 11758	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( B - ( ( B - A ) / 2 ) ) ) e. RR )
24	21	resincld 11758	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( ( B - A ) / 2 ) ) e. RR )
25	23, 24	remulcld 8013	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) e. RR )
26	25	renegcld 8362	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> -u ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) e. RR )
27	22	recoscld 11759	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( B - ( ( B - A ) / 2 ) ) ) e. RR )
28	21	recoscld 11759	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - A ) / 2 ) ) e. RR )
29	27, 28	remulcld 8013	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( cos ` ( B - ( ( B - A ) / 2 ) ) ) x. ( cos ` ( ( B - A ) / 2 ) ) ) e. RR )
30		0red 7983	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 e. RR )
31	1	a1i 9	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 1 e. RR )
32	31	rehalfcld 9190	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 / 2 ) e. RR )
33		simp3 1001	. . . . . . . . . . . . 13 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> A < B )
34	9, 6	posdifd 8514	. . . . . . . . . . . . 13 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( A < B <-> 0 < ( B - A ) ) )
35	33, 34	mpbid 147	. . . . . . . . . . . 12 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( B - A ) )
36		halfpos2 9174	. . . . . . . . . . . . 13 |- ( ( B - A ) e. RR -> ( 0 < ( B - A ) <-> 0 < ( ( B - A ) / 2 ) ) )
37	10, 36	syl 14	. . . . . . . . . . . 12 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 0 < ( B - A ) <-> 0 < ( ( B - A ) / 2 ) ) )
38	35, 37	mpbid 147	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( ( B - A ) / 2 ) )
39		2rp 9683	. . . . . . . . . . . . 13 |- 2 e. RR+
40	39	a1i 9	. . . . . . . . . . . 12 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 2 e. RR+ )
41	2	a1i 9	. . . . . . . . . . . . . 14 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 2 e. RR )
42	1	rexri 8040	. . . . . . . . . . . . . . . 16 |- 1 e. RR*
43	2	rexri 8040	. . . . . . . . . . . . . . . 16 |- 2 e. RR*
44		iccleub 9956	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. RR* /\ 2 e. RR* /\ B e. ( 1 [,] 2 ) ) -> B <_ 2 )
45	42, 43, 44	mp3an12 1338	. . . . . . . . . . . . . . 15 |- ( B e. ( 1 [,] 2 ) -> B <_ 2 )
46	45	3ad2ant2 1021	. . . . . . . . . . . . . 14 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> B <_ 2 )
47		iccgelb 9957	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. RR* /\ 2 e. RR* /\ A e. ( 1 [,] 2 ) ) -> 1 <_ A )
48	42, 43, 47	mp3an12 1338	. . . . . . . . . . . . . . 15 |- ( A e. ( 1 [,] 2 ) -> 1 <_ A )
49	48	3ad2ant1 1020	. . . . . . . . . . . . . 14 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 1 <_ A )
50	6, 31, 41, 9, 46, 49	le2subd 8546	. . . . . . . . . . . . 13 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) <_ ( 2 - 1 ) )
51		2m1e1 9062	. . . . . . . . . . . . 13 |- ( 2 - 1 ) = 1
52	50, 51	breqtrdi 4059	. . . . . . . . . . . 12 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) <_ 1 )
53	10, 31, 40, 52	lediv1dd 9780	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) <_ ( 1 / 2 ) )
54	30, 21, 32, 38, 53	ltletrd 8405	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( 1 / 2 ) )
55		1mhlfehlf 9162	. . . . . . . . . . 11 |- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
56		iccgelb 9957	. . . . . . . . . . . . . 14 |- ( ( 1 e. RR* /\ 2 e. RR* /\ B e. ( 1 [,] 2 ) ) -> 1 <_ B )
57	42, 43, 56	mp3an12 1338	. . . . . . . . . . . . 13 |- ( B e. ( 1 [,] 2 ) -> 1 <_ B )
58	57	3ad2ant2 1021	. . . . . . . . . . . 12 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 1 <_ B )
59	31, 21, 6, 32, 58, 53	le2subd 8546	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 - ( 1 / 2 ) ) <_ ( B - ( ( B - A ) / 2 ) ) )
60	55, 59	eqbrtrrid 4054	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 / 2 ) <_ ( B - ( ( B - A ) / 2 ) ) )
61	30, 32, 22, 54, 60	ltletrd 8405	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( B - ( ( B - A ) / 2 ) ) )
62	30, 21, 38	ltled 8101	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 <_ ( ( B - A ) / 2 ) )
63	6, 30, 41, 21, 46, 62	le2subd 8546	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) <_ ( 2 - 0 ) )
64		2cn 9015	. . . . . . . . . . 11 |- 2 e. CC
65	64	subid1i 8254	. . . . . . . . . 10 |- ( 2 - 0 ) = 2
66	63, 65	breqtrdi 4059	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) <_ 2 )
67		0xr 8029	. . . . . . . . . 10 |- 0 e. RR*
68		elioc2 9961	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ 2 e. RR ) -> ( ( B - ( ( B - A ) / 2 ) ) e. ( 0 (,] 2 ) <-> ( ( B - ( ( B - A ) / 2 ) ) e. RR /\ 0 < ( B - ( ( B - A ) / 2 ) ) /\ ( B - ( ( B - A ) / 2 ) ) <_ 2 ) ) )
69	67, 2, 68	mp2an 426	. . . . . . . . 9 |- ( ( B - ( ( B - A ) / 2 ) ) e. ( 0 (,] 2 ) <-> ( ( B - ( ( B - A ) / 2 ) ) e. RR /\ 0 < ( B - ( ( B - A ) / 2 ) ) /\ ( B - ( ( B - A ) / 2 ) ) <_ 2 ) )
70	22, 61, 66, 69	syl3anbrc 1183	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) e. ( 0 (,] 2 ) )
71		sin02gt0 11798	. . . . . . . 8 |- ( ( B - ( ( B - A ) / 2 ) ) e. ( 0 (,] 2 ) -> 0 < ( sin ` ( B - ( ( B - A ) / 2 ) ) ) )
72	70, 71	syl 14	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( sin ` ( B - ( ( B - A ) / 2 ) ) ) )
73		halfre 9157	. . . . . . . . . . . 12 |- ( 1 / 2 ) e. RR
74		halflt1 9161	. . . . . . . . . . . . 13 |- ( 1 / 2 ) < 1
75		1lt2 9113	. . . . . . . . . . . . 13 |- 1 < 2
76	73, 1, 2	lttri 8087	. . . . . . . . . . . . 13 |- ( ( ( 1 / 2 ) < 1 /\ 1 < 2 ) -> ( 1 / 2 ) < 2 )
77	74, 75, 76	mp2an 426	. . . . . . . . . . . 12 |- ( 1 / 2 ) < 2
78	73, 2, 77	ltleii 8085	. . . . . . . . . . 11 |- ( 1 / 2 ) <_ 2
79	78	a1i 9	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 / 2 ) <_ 2 )
80	21, 32, 41, 53, 79	letrd 8106	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) <_ 2 )
81		elioc2 9961	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ 2 e. RR ) -> ( ( ( B - A ) / 2 ) e. ( 0 (,] 2 ) <-> ( ( ( B - A ) / 2 ) e. RR /\ 0 < ( ( B - A ) / 2 ) /\ ( ( B - A ) / 2 ) <_ 2 ) ) )
82	67, 2, 81	mp2an 426	. . . . . . . . 9 |- ( ( ( B - A ) / 2 ) e. ( 0 (,] 2 ) <-> ( ( ( B - A ) / 2 ) e. RR /\ 0 < ( ( B - A ) / 2 ) /\ ( ( B - A ) / 2 ) <_ 2 ) )
83	21, 38, 80, 82	syl3anbrc 1183	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) e. ( 0 (,] 2 ) )
84		sin02gt0 11798	. . . . . . . 8 |- ( ( ( B - A ) / 2 ) e. ( 0 (,] 2 ) -> 0 < ( sin ` ( ( B - A ) / 2 ) ) )
85	83, 84	syl 14	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( sin ` ( ( B - A ) / 2 ) ) )
86	23, 24, 72, 85	mulgt0d 8105	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) )
87	25	lt0neg2d 8498	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 0 < ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) <-> -u ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) < 0 ) )
88	86, 87	mpbid 147	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> -u ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) < 0 )
89	26, 30, 25, 88, 86	lttrd 8108	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> -u ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) < ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) )
90	26, 25, 29, 89	ltadd2dd 8404	. . 3 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( ( cos ` ( B - ( ( B - A ) / 2 ) ) ) x. ( cos ` ( ( B - A ) / 2 ) ) ) + -u ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) < ( ( ( cos ` ( B - ( ( B - A ) / 2 ) ) ) x. ( cos ` ( ( B - A ) / 2 ) ) ) + ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) )
91	20, 90	eqbrtrrd 4042	. 2 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( ( cos ` ( B - ( ( B - A ) / 2 ) ) ) x. ( cos ` ( ( B - A ) / 2 ) ) ) - ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) < ( ( ( cos ` ( B - ( ( B - A ) / 2 ) ) ) x. ( cos ` ( ( B - A ) / 2 ) ) ) + ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) )
92	7, 12	npcand 8297	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - ( ( B - A ) / 2 ) ) + ( ( B - A ) / 2 ) ) = B )
93	92	fveq2d 5535	. . 3 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - ( ( B - A ) / 2 ) ) + ( ( B - A ) / 2 ) ) ) = ( cos ` B ) )
94		cosadd 11772	. . . 4 |- ( ( ( B - ( ( B - A ) / 2 ) ) e. CC /\ ( ( B - A ) / 2 ) e. CC ) -> ( cos ` ( ( B - ( ( B - A ) / 2 ) ) + ( ( B - A ) / 2 ) ) ) = ( ( ( cos ` ( B - ( ( B - A ) / 2 ) ) ) x. ( cos ` ( ( B - A ) / 2 ) ) ) - ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) )
95	13, 12, 94	syl2anc 411	. . 3 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - ( ( B - A ) / 2 ) ) + ( ( B - A ) / 2 ) ) ) = ( ( ( cos ` ( B - ( ( B - A ) / 2 ) ) ) x. ( cos ` ( ( B - A ) / 2 ) ) ) - ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) )
96	93, 95	eqtr3d 2224	. 2 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` B ) = ( ( ( cos ` ( B - ( ( B - A ) / 2 ) ) ) x. ( cos ` ( ( B - A ) / 2 ) ) ) - ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) )
97	7, 12, 12	subsub4d 8324	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - ( ( B - A ) / 2 ) ) - ( ( B - A ) / 2 ) ) = ( B - ( ( ( B - A ) / 2 ) + ( ( B - A ) / 2 ) ) ) )
98	11	2halvesd 9189	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( ( B - A ) / 2 ) + ( ( B - A ) / 2 ) ) = ( B - A ) )
99	98	oveq2d 5908	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( ( B - A ) / 2 ) + ( ( B - A ) / 2 ) ) ) = ( B - ( B - A ) ) )
100	9	recnd 8011	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> A e. CC )
101	7, 100	nncand 8298	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( B - A ) ) = A )
102	97, 99, 101	3eqtrd 2226	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - ( ( B - A ) / 2 ) ) - ( ( B - A ) / 2 ) ) = A )
103	102	fveq2d 5535	. . 3 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - ( ( B - A ) / 2 ) ) - ( ( B - A ) / 2 ) ) ) = ( cos ` A ) )
104		cossub 11776	. . . 4 |- ( ( ( B - ( ( B - A ) / 2 ) ) e. CC /\ ( ( B - A ) / 2 ) e. CC ) -> ( cos ` ( ( B - ( ( B - A ) / 2 ) ) - ( ( B - A ) / 2 ) ) ) = ( ( ( cos ` ( B - ( ( B - A ) / 2 ) ) ) x. ( cos ` ( ( B - A ) / 2 ) ) ) + ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) )
105	13, 12, 104	syl2anc 411	. . 3 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - ( ( B - A ) / 2 ) ) - ( ( B - A ) / 2 ) ) ) = ( ( ( cos ` ( B - ( ( B - A ) / 2 ) ) ) x. ( cos ` ( ( B - A ) / 2 ) ) ) + ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) )
106	103, 105	eqtr3d 2224	. 2 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` A ) = ( ( ( cos ` ( B - ( ( B - A ) / 2 ) ) ) x. ( cos ` ( ( B - A ) / 2 ) ) ) + ( ( sin ` ( B - ( ( B - A ) / 2 ) ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) )
107	91, 96, 106	3brtr4d 4050	1 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` B ) < ( cos ` A ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2160   C_ wss 3144   class class class wbr 4018   ` cfv 5232  (class class class)co 5892   CCcc 7834   RRcr 7835   0cc0 7836   1c1 7837   + caddc 7839   x. cmul 7841   RR*cxr 8016   < clt 8017   <_ cle 8018   - cmin 8153   -ucneg 8154   / cdiv 8654   2c2 8995   RR+crp 9678   (,]cioc 9914   [,]cicc 9916   sincsin 11679   cosccos 11680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-mulrcl 7935  ax-addcom 7936  ax-mulcom 7937  ax-addass 7938  ax-mulass 7939  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-1rid 7943  ax-0id 7944  ax-rnegex 7945  ax-precex 7946  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-apti 7951  ax-pre-ltadd 7952  ax-pre-mulgt0 7953  ax-pre-mulext 7954  ax-arch 7955  ax-caucvg 7956

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-disj 3996  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-isom 5241  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-frec 6411  df-1o 6436  df-oadd 6440  df-er 6554  df-en 6762  df-dom 6763  df-fin 6764  df-sup 7008  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-reap 8557  df-ap 8564  df-div 8655  df-inn 8945  df-2 9003  df-3 9004  df-4 9005  df-5 9006  df-6 9007  df-7 9008  df-8 9009  df-n0 9202  df-z 9279  df-uz 9554  df-q 9645  df-rp 9679  df-ioc 9918  df-ico 9919  df-icc 9920  df-fz 10034  df-fzo 10168  df-seqfrec 10472  df-exp 10546  df-fac 10733  df-bc 10755  df-ihash 10783  df-shft 10851  df-cj 10878  df-re 10879  df-im 10880  df-rsqrt 11034  df-abs 11035  df-clim 11314  df-sumdc 11389  df-ef 11683  df-sin 11685  df-cos 11686

This theorem is referenced by:  cosz12  14638