Theorem cos12dec 11730

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 7919	. . . . . . . . . . 11 |- 1 e. RR
2		2re 8948	. . . . . . . . . . 11 |- 2 e. RR
3		iccssre 9912	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ 2 e. RR ) -> ( 1 [,] 2 ) C_ RR )
4	1, 2, 3	mp2an 424	. . . . . . . . . 10 |- ( 1 [,] 2 ) C_ RR
5	4	sseli 3143	. . . . . . . . 9 |- ( B e. ( 1 [,] 2 ) -> B e. RR )
6	5	3ad2ant2 1014	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> B e. RR )
7	6	recnd 7948	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> B e. CC )
8	4	sseli 3143	. . . . . . . . . . 11 |- ( A e. ( 1 [,] 2 ) -> A e. RR )
9	8	3ad2ant1 1013	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> A e. RR )
10	6, 9	resubcld 8300	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) e. RR )
11	10	recnd 7948	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) e. CC )
12	11	halfcld 9122	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) e. CC )
13	7, 12	subcld 8230	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) e. CC )
14	13	coscld 11674	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( B - ( ( B - A ) / 2 ) ) ) e. CC )
15	12	coscld 11674	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - A ) / 2 ) ) e. CC )
16	14, 15	mulcld 7940	. . . 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 11673	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( B - ( ( B - A ) / 2 ) ) ) e. CC )
18	12	sincld 11673	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( ( B - A ) / 2 ) ) e. CC )
19	17, 18	mulcld 7940	. . . 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 8236	. . 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 9124	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) e. RR )
22	6, 21	resubcld 8300	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) e. RR )
23	22	resincld 11686	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( B - ( ( B - A ) / 2 ) ) ) e. RR )
24	21	resincld 11686	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( ( B - A ) / 2 ) ) e. RR )
25	23, 24	remulcld 7950	. . . . 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 8299	. . . 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 11687	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( B - ( ( B - A ) / 2 ) ) ) e. RR )
28	21	recoscld 11687	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - A ) / 2 ) ) e. RR )
29	27, 28	remulcld 7950	. . . 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 7921	. . . . 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 9124	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 / 2 ) e. RR )
33		simp3 994	. . . . . . . . . . . . 13 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> A < B )
34	9, 6	posdifd 8451	. . . . . . . . . . . . 13 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( A < B <-> 0 < ( B - A ) ) )
35	33, 34	mpbid 146	. . . . . . . . . . . 12 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( B - A ) )
36		halfpos2 9108	. . . . . . . . . . . . 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 146	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( ( B - A ) / 2 ) )
39		2rp 9615	. . . . . . . . . . . . 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 7977	. . . . . . . . . . . . . . . 16 |- 1 e. RR*
43	2	rexri 7977	. . . . . . . . . . . . . . . 16 |- 2 e. RR*
44		iccleub 9888	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. RR* /\ 2 e. RR* /\ B e. ( 1 [,] 2 ) ) -> B <_ 2 )
45	42, 43, 44	mp3an12 1322	. . . . . . . . . . . . . . 15 |- ( B e. ( 1 [,] 2 ) -> B <_ 2 )
46	45	3ad2ant2 1014	. . . . . . . . . . . . . 14 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> B <_ 2 )
47		iccgelb 9889	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. RR* /\ 2 e. RR* /\ A e. ( 1 [,] 2 ) ) -> 1 <_ A )
48	42, 43, 47	mp3an12 1322	. . . . . . . . . . . . . . 15 |- ( A e. ( 1 [,] 2 ) -> 1 <_ A )
49	48	3ad2ant1 1013	. . . . . . . . . . . . . 14 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 1 <_ A )
50	6, 31, 41, 9, 46, 49	le2subd 8483	. . . . . . . . . . . . 13 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) <_ ( 2 - 1 ) )
51		2m1e1 8996	. . . . . . . . . . . . 13 |- ( 2 - 1 ) = 1
52	50, 51	breqtrdi 4030	. . . . . . . . . . . 12 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) <_ 1 )
53	10, 31, 40, 52	lediv1dd 9712	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) <_ ( 1 / 2 ) )
54	30, 21, 32, 38, 53	ltletrd 8342	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( 1 / 2 ) )
55		1mhlfehlf 9096	. . . . . . . . . . 11 |- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
56		iccgelb 9889	. . . . . . . . . . . . . 14 |- ( ( 1 e. RR* /\ 2 e. RR* /\ B e. ( 1 [,] 2 ) ) -> 1 <_ B )
57	42, 43, 56	mp3an12 1322	. . . . . . . . . . . . 13 |- ( B e. ( 1 [,] 2 ) -> 1 <_ B )
58	57	3ad2ant2 1014	. . . . . . . . . . . 12 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 1 <_ B )
59	31, 21, 6, 32, 58, 53	le2subd 8483	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 - ( 1 / 2 ) ) <_ ( B - ( ( B - A ) / 2 ) ) )
60	55, 59	eqbrtrrid 4025	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 / 2 ) <_ ( B - ( ( B - A ) / 2 ) ) )
61	30, 32, 22, 54, 60	ltletrd 8342	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( B - ( ( B - A ) / 2 ) ) )
62	30, 21, 38	ltled 8038	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 <_ ( ( B - A ) / 2 ) )
63	6, 30, 41, 21, 46, 62	le2subd 8483	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) <_ ( 2 - 0 ) )
64		2cn 8949	. . . . . . . . . . 11 |- 2 e. CC
65	64	subid1i 8191	. . . . . . . . . 10 |- ( 2 - 0 ) = 2
66	63, 65	breqtrdi 4030	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) <_ 2 )
67		0xr 7966	. . . . . . . . . 10 |- 0 e. RR*
68		elioc2 9893	. . . . . . . . . 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 424	. . . . . . . . 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 1176	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) e. ( 0 (,] 2 ) )
71		sin02gt0 11726	. . . . . . . 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 9091	. . . . . . . . . . . 12 |- ( 1 / 2 ) e. RR
74		halflt1 9095	. . . . . . . . . . . . 13 |- ( 1 / 2 ) < 1
75		1lt2 9047	. . . . . . . . . . . . 13 |- 1 < 2
76	73, 1, 2	lttri 8024	. . . . . . . . . . . . 13 |- ( ( ( 1 / 2 ) < 1 /\ 1 < 2 ) -> ( 1 / 2 ) < 2 )
77	74, 75, 76	mp2an 424	. . . . . . . . . . . 12 |- ( 1 / 2 ) < 2
78	73, 2, 77	ltleii 8022	. . . . . . . . . . 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 8043	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) <_ 2 )
81		elioc2 9893	. . . . . . . . . 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 424	. . . . . . . . 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 1176	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) e. ( 0 (,] 2 ) )
84		sin02gt0 11726	. . . . . . . 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 8042	. . . . . 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 8435	. . . . . 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 146	. . . . 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 8045	. . . 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 8341	. . 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 4013	. 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 8234	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - ( ( B - A ) / 2 ) ) + ( ( B - A ) / 2 ) ) = B )
93	92	fveq2d 5500	. . 3 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - ( ( B - A ) / 2 ) ) + ( ( B - A ) / 2 ) ) ) = ( cos ` B ) )
94		cosadd 11700	. . . 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 409	. . 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 2205	. 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 8261	. . . . 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 9123	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( ( B - A ) / 2 ) + ( ( B - A ) / 2 ) ) = ( B - A ) )
99	98	oveq2d 5869	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( ( B - A ) / 2 ) + ( ( B - A ) / 2 ) ) ) = ( B - ( B - A ) ) )
100	9	recnd 7948	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> A e. CC )
101	7, 100	nncand 8235	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( B - A ) ) = A )
102	97, 99, 101	3eqtrd 2207	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - ( ( B - A ) / 2 ) ) - ( ( B - A ) / 2 ) ) = A )
103	102	fveq2d 5500	. . 3 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - ( ( B - A ) / 2 ) ) - ( ( B - A ) / 2 ) ) ) = ( cos ` A ) )
104		cossub 11704	. . . 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 409	. . 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 2205	. 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 4021	1 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` B ) < ( cos ` A ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   C_ wss 3121   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   RR*cxr 7953   < clt 7954   <_ cle 7955   - cmin 8090   -ucneg 8091   / cdiv 8589   2c2 8929   RR+crp 9610   (,]cioc 9846   [,]cicc 9848   sincsin 11607   cosccos 11608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-disj 3967  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-5 8940  df-6 8941  df-7 8942  df-8 8943  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-ioc 9850  df-ico 9851  df-icc 9852  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-bc 10682  df-ihash 10710  df-shft 10779  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-ef 11611  df-sin 11613  df-cos 11614

This theorem is referenced by:  cosz12  13495