Theorem cos12dec 11914

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 8020	. . . . . . . . . . 11 |- 1 e. RR
2		2re 9054	. . . . . . . . . . 11 |- 2 e. RR
3		iccssre 10024	. . . . . . . . . . 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 3176	. . . . . . . . 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 8050	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> B e. CC )
8	4	sseli 3176	. . . . . . . . . . 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 8402	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) e. RR )
11	10	recnd 8050	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) e. CC )
12	11	halfcld 9230	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) e. CC )
13	7, 12	subcld 8332	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) e. CC )
14	13	coscld 11857	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( B - ( ( B - A ) / 2 ) ) ) e. CC )
15	12	coscld 11857	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - A ) / 2 ) ) e. CC )
16	14, 15	mulcld 8042	. . . 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 11856	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( B - ( ( B - A ) / 2 ) ) ) e. CC )
18	12	sincld 11856	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( ( B - A ) / 2 ) ) e. CC )
19	17, 18	mulcld 8042	. . . 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 8338	. . 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 9232	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) e. RR )
22	6, 21	resubcld 8402	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) e. RR )
23	22	resincld 11869	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( B - ( ( B - A ) / 2 ) ) ) e. RR )
24	21	resincld 11869	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( ( B - A ) / 2 ) ) e. RR )
25	23, 24	remulcld 8052	. . . . 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 8401	. . . 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 11870	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( B - ( ( B - A ) / 2 ) ) ) e. RR )
28	21	recoscld 11870	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - A ) / 2 ) ) e. RR )
29	27, 28	remulcld 8052	. . . 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 8022	. . . . 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 9232	. . . . . . . . . 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 8553	. . . . . . . . . . . . 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 9215	. . . . . . . . . . . . 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 9727	. . . . . . . . . . . . 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 8079	. . . . . . . . . . . . . . . 16 |- 1 e. RR*
43	2	rexri 8079	. . . . . . . . . . . . . . . 16 |- 2 e. RR*
44		iccleub 10000	. . . . . . . . . . . . . . . 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 10001	. . . . . . . . . . . . . . . 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 8585	. . . . . . . . . . . . 13 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) <_ ( 2 - 1 ) )
51		2m1e1 9102	. . . . . . . . . . . . 13 |- ( 2 - 1 ) = 1
52	50, 51	breqtrdi 4071	. . . . . . . . . . . 12 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) <_ 1 )
53	10, 31, 40, 52	lediv1dd 9824	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) <_ ( 1 / 2 ) )
54	30, 21, 32, 38, 53	ltletrd 8444	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( 1 / 2 ) )
55		1mhlfehlf 9203	. . . . . . . . . . 11 |- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
56		iccgelb 10001	. . . . . . . . . . . . . 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 8585	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 - ( 1 / 2 ) ) <_ ( B - ( ( B - A ) / 2 ) ) )
60	55, 59	eqbrtrrid 4066	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 / 2 ) <_ ( B - ( ( B - A ) / 2 ) ) )
61	30, 32, 22, 54, 60	ltletrd 8444	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( B - ( ( B - A ) / 2 ) ) )
62	30, 21, 38	ltled 8140	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 <_ ( ( B - A ) / 2 ) )
63	6, 30, 41, 21, 46, 62	le2subd 8585	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) <_ ( 2 - 0 ) )
64		2cn 9055	. . . . . . . . . . 11 |- 2 e. CC
65	64	subid1i 8293	. . . . . . . . . 10 |- ( 2 - 0 ) = 2
66	63, 65	breqtrdi 4071	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) <_ 2 )
67		0xr 8068	. . . . . . . . . 10 |- 0 e. RR*
68		elioc2 10005	. . . . . . . . . 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 11910	. . . . . . . 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 9198	. . . . . . . . . . . 12 |- ( 1 / 2 ) e. RR
74		halflt1 9202	. . . . . . . . . . . . 13 |- ( 1 / 2 ) < 1
75		1lt2 9154	. . . . . . . . . . . . 13 |- 1 < 2
76	73, 1, 2	lttri 8126	. . . . . . . . . . . . 13 |- ( ( ( 1 / 2 ) < 1 /\ 1 < 2 ) -> ( 1 / 2 ) < 2 )
77	74, 75, 76	mp2an 426	. . . . . . . . . . . 12 |- ( 1 / 2 ) < 2
78	73, 2, 77	ltleii 8124	. . . . . . . . . . 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 8145	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) <_ 2 )
81		elioc2 10005	. . . . . . . . . 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 11910	. . . . . . . 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 8144	. . . . . 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 8537	. . . . . 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 8147	. . . 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 8443	. . 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 4054	. 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 8336	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - ( ( B - A ) / 2 ) ) + ( ( B - A ) / 2 ) ) = B )
93	92	fveq2d 5559	. . 3 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - ( ( B - A ) / 2 ) ) + ( ( B - A ) / 2 ) ) ) = ( cos ` B ) )
94		cosadd 11883	. . . 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 2228	. 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 8363	. . . . 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 9231	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( ( B - A ) / 2 ) + ( ( B - A ) / 2 ) ) = ( B - A ) )
99	98	oveq2d 5935	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( ( B - A ) / 2 ) + ( ( B - A ) / 2 ) ) ) = ( B - ( B - A ) ) )
100	9	recnd 8050	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> A e. CC )
101	7, 100	nncand 8337	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( B - A ) ) = A )
102	97, 99, 101	3eqtrd 2230	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - ( ( B - A ) / 2 ) ) - ( ( B - A ) / 2 ) ) = A )
103	102	fveq2d 5559	. . 3 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - ( ( B - A ) / 2 ) ) - ( ( B - A ) / 2 ) ) ) = ( cos ` A ) )
104		cossub 11887	. . . 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 2228	. 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 4062	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 2164   C_ wss 3154   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   CCcc 7872   RRcr 7873   0cc0 7874   1c1 7875   + caddc 7877   x. cmul 7879   RR*cxr 8055   < clt 8056   <_ cle 8057   - cmin 8192   -ucneg 8193   / cdiv 8693   2c2 9035   RR+crp 9722   (,]cioc 9958   [,]cicc 9960   sincsin 11790   cosccos 11791

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-disj 4008  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-oadd 6475  df-er 6589  df-en 6797  df-dom 6798  df-fin 6799  df-sup 7045  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-ioc 9962  df-ico 9963  df-icc 9964  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-exp 10613  df-fac 10800  df-bc 10822  df-ihash 10850  df-shft 10962  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-sumdc 11500  df-ef 11794  df-sin 11796  df-cos 11797

This theorem is referenced by:  cosz12  14956