Theorem cos12dec 12274

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 8141	. . . . . . . . . . 11 |- 1 e. RR
2		2re 9176	. . . . . . . . . . 11 |- 2 e. RR
3		iccssre 10147	. . . . . . . . . . 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 3220	. . . . . . . . 9 |- ( B e. ( 1 [,] 2 ) -> B e. RR )
6	5	3ad2ant2 1043	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> B e. RR )
7	6	recnd 8171	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> B e. CC )
8	4	sseli 3220	. . . . . . . . . . 11 |- ( A e. ( 1 [,] 2 ) -> A e. RR )
9	8	3ad2ant1 1042	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> A e. RR )
10	6, 9	resubcld 8523	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) e. RR )
11	10	recnd 8171	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) e. CC )
12	11	halfcld 9352	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) e. CC )
13	7, 12	subcld 8453	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) e. CC )
14	13	coscld 12217	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( B - ( ( B - A ) / 2 ) ) ) e. CC )
15	12	coscld 12217	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - A ) / 2 ) ) e. CC )
16	14, 15	mulcld 8163	. . . 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 12216	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( B - ( ( B - A ) / 2 ) ) ) e. CC )
18	12	sincld 12216	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( ( B - A ) / 2 ) ) e. CC )
19	17, 18	mulcld 8163	. . . 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 8459	. . 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 9354	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) e. RR )
22	6, 21	resubcld 8523	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) e. RR )
23	22	resincld 12229	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( B - ( ( B - A ) / 2 ) ) ) e. RR )
24	21	resincld 12229	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( ( B - A ) / 2 ) ) e. RR )
25	23, 24	remulcld 8173	. . . . 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 8522	. . . 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 12230	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( B - ( ( B - A ) / 2 ) ) ) e. RR )
28	21	recoscld 12230	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - A ) / 2 ) ) e. RR )
29	27, 28	remulcld 8173	. . . 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 8143	. . . . 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 9354	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 / 2 ) e. RR )
33		simp3 1023	. . . . . . . . . . . . 13 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> A < B )
34	9, 6	posdifd 8675	. . . . . . . . . . . . 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 9337	. . . . . . . . . . . . 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 9850	. . . . . . . . . . . . 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 8200	. . . . . . . . . . . . . . . 16 |- 1 e. RR*
43	2	rexri 8200	. . . . . . . . . . . . . . . 16 |- 2 e. RR*
44		iccleub 10123	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. RR* /\ 2 e. RR* /\ B e. ( 1 [,] 2 ) ) -> B <_ 2 )
45	42, 43, 44	mp3an12 1361	. . . . . . . . . . . . . . 15 |- ( B e. ( 1 [,] 2 ) -> B <_ 2 )
46	45	3ad2ant2 1043	. . . . . . . . . . . . . 14 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> B <_ 2 )
47		iccgelb 10124	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. RR* /\ 2 e. RR* /\ A e. ( 1 [,] 2 ) ) -> 1 <_ A )
48	42, 43, 47	mp3an12 1361	. . . . . . . . . . . . . . 15 |- ( A e. ( 1 [,] 2 ) -> 1 <_ A )
49	48	3ad2ant1 1042	. . . . . . . . . . . . . 14 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 1 <_ A )
50	6, 31, 41, 9, 46, 49	le2subd 8707	. . . . . . . . . . . . 13 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) <_ ( 2 - 1 ) )
51		2m1e1 9224	. . . . . . . . . . . . 13 |- ( 2 - 1 ) = 1
52	50, 51	breqtrdi 4123	. . . . . . . . . . . 12 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) <_ 1 )
53	10, 31, 40, 52	lediv1dd 9947	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) <_ ( 1 / 2 ) )
54	30, 21, 32, 38, 53	ltletrd 8566	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( 1 / 2 ) )
55		1mhlfehlf 9325	. . . . . . . . . . 11 |- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
56		iccgelb 10124	. . . . . . . . . . . . . 14 |- ( ( 1 e. RR* /\ 2 e. RR* /\ B e. ( 1 [,] 2 ) ) -> 1 <_ B )
57	42, 43, 56	mp3an12 1361	. . . . . . . . . . . . 13 |- ( B e. ( 1 [,] 2 ) -> 1 <_ B )
58	57	3ad2ant2 1043	. . . . . . . . . . . 12 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 1 <_ B )
59	31, 21, 6, 32, 58, 53	le2subd 8707	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 - ( 1 / 2 ) ) <_ ( B - ( ( B - A ) / 2 ) ) )
60	55, 59	eqbrtrrid 4118	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 / 2 ) <_ ( B - ( ( B - A ) / 2 ) ) )
61	30, 32, 22, 54, 60	ltletrd 8566	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( B - ( ( B - A ) / 2 ) ) )
62	30, 21, 38	ltled 8261	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 <_ ( ( B - A ) / 2 ) )
63	6, 30, 41, 21, 46, 62	le2subd 8707	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) <_ ( 2 - 0 ) )
64		2cn 9177	. . . . . . . . . . 11 |- 2 e. CC
65	64	subid1i 8414	. . . . . . . . . 10 |- ( 2 - 0 ) = 2
66	63, 65	breqtrdi 4123	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) <_ 2 )
67		0xr 8189	. . . . . . . . . 10 |- 0 e. RR*
68		elioc2 10128	. . . . . . . . . 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 1205	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) e. ( 0 (,] 2 ) )
71		sin02gt0 12270	. . . . . . . 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 9320	. . . . . . . . . . . 12 |- ( 1 / 2 ) e. RR
74		halflt1 9324	. . . . . . . . . . . . 13 |- ( 1 / 2 ) < 1
75		1lt2 9276	. . . . . . . . . . . . 13 |- 1 < 2
76	73, 1, 2	lttri 8247	. . . . . . . . . . . . 13 |- ( ( ( 1 / 2 ) < 1 /\ 1 < 2 ) -> ( 1 / 2 ) < 2 )
77	74, 75, 76	mp2an 426	. . . . . . . . . . . 12 |- ( 1 / 2 ) < 2
78	73, 2, 77	ltleii 8245	. . . . . . . . . . 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 8266	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) <_ 2 )
81		elioc2 10128	. . . . . . . . . 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 1205	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) e. ( 0 (,] 2 ) )
84		sin02gt0 12270	. . . . . . . 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 8265	. . . . . 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 8659	. . . . . 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 8268	. . . 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 8565	. . 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 4106	. 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 8457	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - ( ( B - A ) / 2 ) ) + ( ( B - A ) / 2 ) ) = B )
93	92	fveq2d 5630	. . 3 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - ( ( B - A ) / 2 ) ) + ( ( B - A ) / 2 ) ) ) = ( cos ` B ) )
94		cosadd 12243	. . . 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 2264	. 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 8484	. . . . 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 9353	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( ( B - A ) / 2 ) + ( ( B - A ) / 2 ) ) = ( B - A ) )
99	98	oveq2d 6016	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( ( B - A ) / 2 ) + ( ( B - A ) / 2 ) ) ) = ( B - ( B - A ) ) )
100	9	recnd 8171	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> A e. CC )
101	7, 100	nncand 8458	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( B - A ) ) = A )
102	97, 99, 101	3eqtrd 2266	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - ( ( B - A ) / 2 ) ) - ( ( B - A ) / 2 ) ) = A )
103	102	fveq2d 5630	. . 3 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - ( ( B - A ) / 2 ) ) - ( ( B - A ) / 2 ) ) ) = ( cos ` A ) )
104		cossub 12247	. . . 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 2264	. 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 4114	1 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` B ) < ( cos ` A ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   C_ wss 3197   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   CCcc 7993   RRcr 7994   0cc0 7995   1c1 7996   + caddc 7998   x. cmul 8000   RR*cxr 8176   < clt 8177   <_ cle 8178   - cmin 8313   -ucneg 8314   / cdiv 8815   2c2 9157   RR+crp 9845   (,]cioc 10081   [,]cicc 10083   sincsin 12150   cosccos 12151

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-disj 4059  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-frec 6535  df-1o 6560  df-oadd 6564  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-sup 7147  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-5 9168  df-6 9169  df-7 9170  df-8 9171  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-ioc 10085  df-ico 10086  df-icc 10087  df-fz 10201  df-fzo 10335  df-seqfrec 10665  df-exp 10756  df-fac 10943  df-bc 10965  df-ihash 10993  df-shft 11321  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-clim 11785  df-sumdc 11860  df-ef 12154  df-sin 12156  df-cos 12157

This theorem is referenced by:  cosz12  15448