Theorem cos12dec 11759

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 7947	. . . . . . . . . . 11 |- 1 e. RR
2		2re 8978	. . . . . . . . . . 11 |- 2 e. RR
3		iccssre 9942	. . . . . . . . . . 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 3151	. . . . . . . . 9 |- ( B e. ( 1 [,] 2 ) -> B e. RR )
6	5	3ad2ant2 1019	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> B e. RR )
7	6	recnd 7976	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> B e. CC )
8	4	sseli 3151	. . . . . . . . . . 11 |- ( A e. ( 1 [,] 2 ) -> A e. RR )
9	8	3ad2ant1 1018	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> A e. RR )
10	6, 9	resubcld 8328	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) e. RR )
11	10	recnd 7976	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) e. CC )
12	11	halfcld 9152	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) e. CC )
13	7, 12	subcld 8258	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) e. CC )
14	13	coscld 11703	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( B - ( ( B - A ) / 2 ) ) ) e. CC )
15	12	coscld 11703	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - A ) / 2 ) ) e. CC )
16	14, 15	mulcld 7968	. . . 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 11702	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( B - ( ( B - A ) / 2 ) ) ) e. CC )
18	12	sincld 11702	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( ( B - A ) / 2 ) ) e. CC )
19	17, 18	mulcld 7968	. . . 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 8264	. . 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 9154	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) e. RR )
22	6, 21	resubcld 8328	. . . . . . 7 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) e. RR )
23	22	resincld 11715	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( B - ( ( B - A ) / 2 ) ) ) e. RR )
24	21	resincld 11715	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( sin ` ( ( B - A ) / 2 ) ) e. RR )
25	23, 24	remulcld 7978	. . . . 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 8327	. . . 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 11716	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( B - ( ( B - A ) / 2 ) ) ) e. RR )
28	21	recoscld 11716	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - A ) / 2 ) ) e. RR )
29	27, 28	remulcld 7978	. . . 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 7949	. . . . 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 9154	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 / 2 ) e. RR )
33		simp3 999	. . . . . . . . . . . . 13 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> A < B )
34	9, 6	posdifd 8479	. . . . . . . . . . . . 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 9138	. . . . . . . . . . . . 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 9645	. . . . . . . . . . . . 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 8005	. . . . . . . . . . . . . . . 16 |- 1 e. RR*
43	2	rexri 8005	. . . . . . . . . . . . . . . 16 |- 2 e. RR*
44		iccleub 9918	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. RR* /\ 2 e. RR* /\ B e. ( 1 [,] 2 ) ) -> B <_ 2 )
45	42, 43, 44	mp3an12 1327	. . . . . . . . . . . . . . 15 |- ( B e. ( 1 [,] 2 ) -> B <_ 2 )
46	45	3ad2ant2 1019	. . . . . . . . . . . . . 14 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> B <_ 2 )
47		iccgelb 9919	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. RR* /\ 2 e. RR* /\ A e. ( 1 [,] 2 ) ) -> 1 <_ A )
48	42, 43, 47	mp3an12 1327	. . . . . . . . . . . . . . 15 |- ( A e. ( 1 [,] 2 ) -> 1 <_ A )
49	48	3ad2ant1 1018	. . . . . . . . . . . . . 14 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 1 <_ A )
50	6, 31, 41, 9, 46, 49	le2subd 8511	. . . . . . . . . . . . 13 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) <_ ( 2 - 1 ) )
51		2m1e1 9026	. . . . . . . . . . . . 13 |- ( 2 - 1 ) = 1
52	50, 51	breqtrdi 4041	. . . . . . . . . . . 12 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - A ) <_ 1 )
53	10, 31, 40, 52	lediv1dd 9742	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) <_ ( 1 / 2 ) )
54	30, 21, 32, 38, 53	ltletrd 8370	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( 1 / 2 ) )
55		1mhlfehlf 9126	. . . . . . . . . . 11 |- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
56		iccgelb 9919	. . . . . . . . . . . . . 14 |- ( ( 1 e. RR* /\ 2 e. RR* /\ B e. ( 1 [,] 2 ) ) -> 1 <_ B )
57	42, 43, 56	mp3an12 1327	. . . . . . . . . . . . 13 |- ( B e. ( 1 [,] 2 ) -> 1 <_ B )
58	57	3ad2ant2 1019	. . . . . . . . . . . 12 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 1 <_ B )
59	31, 21, 6, 32, 58, 53	le2subd 8511	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 - ( 1 / 2 ) ) <_ ( B - ( ( B - A ) / 2 ) ) )
60	55, 59	eqbrtrrid 4036	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( 1 / 2 ) <_ ( B - ( ( B - A ) / 2 ) ) )
61	30, 32, 22, 54, 60	ltletrd 8370	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 < ( B - ( ( B - A ) / 2 ) ) )
62	30, 21, 38	ltled 8066	. . . . . . . . . . 11 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> 0 <_ ( ( B - A ) / 2 ) )
63	6, 30, 41, 21, 46, 62	le2subd 8511	. . . . . . . . . 10 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) <_ ( 2 - 0 ) )
64		2cn 8979	. . . . . . . . . . 11 |- 2 e. CC
65	64	subid1i 8219	. . . . . . . . . 10 |- ( 2 - 0 ) = 2
66	63, 65	breqtrdi 4041	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) <_ 2 )
67		0xr 7994	. . . . . . . . . 10 |- 0 e. RR*
68		elioc2 9923	. . . . . . . . . 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 1181	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( B - A ) / 2 ) ) e. ( 0 (,] 2 ) )
71		sin02gt0 11755	. . . . . . . 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 9121	. . . . . . . . . . . 12 |- ( 1 / 2 ) e. RR
74		halflt1 9125	. . . . . . . . . . . . 13 |- ( 1 / 2 ) < 1
75		1lt2 9077	. . . . . . . . . . . . 13 |- 1 < 2
76	73, 1, 2	lttri 8052	. . . . . . . . . . . . 13 |- ( ( ( 1 / 2 ) < 1 /\ 1 < 2 ) -> ( 1 / 2 ) < 2 )
77	74, 75, 76	mp2an 426	. . . . . . . . . . . 12 |- ( 1 / 2 ) < 2
78	73, 2, 77	ltleii 8050	. . . . . . . . . . 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 8071	. . . . . . . . 9 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) <_ 2 )
81		elioc2 9923	. . . . . . . . . 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 1181	. . . . . . . 8 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - A ) / 2 ) e. ( 0 (,] 2 ) )
84		sin02gt0 11755	. . . . . . . 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 8070	. . . . . 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 8463	. . . . . 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 8073	. . . 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 8369	. . 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 4024	. 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 8262	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - ( ( B - A ) / 2 ) ) + ( ( B - A ) / 2 ) ) = B )
93	92	fveq2d 5515	. . 3 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - ( ( B - A ) / 2 ) ) + ( ( B - A ) / 2 ) ) ) = ( cos ` B ) )
94		cosadd 11729	. . . 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 2212	. 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 8289	. . . . 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 9153	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( ( B - A ) / 2 ) + ( ( B - A ) / 2 ) ) = ( B - A ) )
99	98	oveq2d 5885	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( ( ( B - A ) / 2 ) + ( ( B - A ) / 2 ) ) ) = ( B - ( B - A ) ) )
100	9	recnd 7976	. . . . . 6 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> A e. CC )
101	7, 100	nncand 8263	. . . . 5 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( B - ( B - A ) ) = A )
102	97, 99, 101	3eqtrd 2214	. . . 4 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( ( B - ( ( B - A ) / 2 ) ) - ( ( B - A ) / 2 ) ) = A )
103	102	fveq2d 5515	. . 3 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` ( ( B - ( ( B - A ) / 2 ) ) - ( ( B - A ) / 2 ) ) ) = ( cos ` A ) )
104		cossub 11733	. . . 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 2212	. 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 4032	1 |- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` B ) < ( cos ` A ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   C_ wss 3129   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803   + caddc 7805   x. cmul 7807   RR*cxr 7981   < clt 7982   <_ cle 7983   - cmin 8118   -ucneg 8119   / cdiv 8618   2c2 8959   RR+crp 9640   (,]cioc 9876   [,]cicc 9878   sincsin 11636   cosccos 11637

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-disj 3978  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-sup 6977  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-7 8972  df-8 8973  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-ioc 9880  df-ico 9881  df-icc 9882  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-fac 10690  df-bc 10712  df-ihash 10740  df-shft 10808  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-sumdc 11346  df-ef 11640  df-sin 11642  df-cos 11643

This theorem is referenced by:  cosz12  13868