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Theorem cos12dec 11481
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
StepHypRef Expression
1 1re 7772 . . . . . . . . . . 11  |-  1  e.  RR
2 2re 8797 . . . . . . . . . . 11  |-  2  e.  RR
3 iccssre 9745 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  2  e.  RR )  ->  ( 1 [,] 2
)  C_  RR )
41, 2, 3mp2an 422 . . . . . . . . . 10  |-  ( 1 [,] 2 )  C_  RR
54sseli 3093 . . . . . . . . 9  |-  ( B  e.  ( 1 [,] 2 )  ->  B  e.  RR )
653ad2ant2 1003 . . . . . . . 8  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  ->  B  e.  RR )
76recnd 7801 . . . . . . 7  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  ->  B  e.  CC )
84sseli 3093 . . . . . . . . . . 11  |-  ( A  e.  ( 1 [,] 2 )  ->  A  e.  RR )
983ad2ant1 1002 . . . . . . . . . 10  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  ->  A  e.  RR )
106, 9resubcld 8150 . . . . . . . . 9  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( B  -  A
)  e.  RR )
1110recnd 7801 . . . . . . . 8  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( B  -  A
)  e.  CC )
1211halfcld 8971 . . . . . . 7  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( ( B  -  A )  /  2
)  e.  CC )
137, 12subcld 8080 . . . . . 6  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( B  -  (
( B  -  A
)  /  2 ) )  e.  CC )
1413coscld 11425 . . . . 5  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( cos `  ( B  -  ( ( B  -  A )  /  2 ) ) )  e.  CC )
1512coscld 11425 . . . . 5  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( cos `  (
( B  -  A
)  /  2 ) )  e.  CC )
1614, 15mulcld 7793 . . . 4  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( ( cos `  ( B  -  ( ( B  -  A )  /  2 ) ) )  x.  ( cos `  ( ( B  -  A )  /  2
) ) )  e.  CC )
1713sincld 11424 . . . . 5  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( sin `  ( B  -  ( ( B  -  A )  /  2 ) ) )  e.  CC )
1812sincld 11424 . . . . 5  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( sin `  (
( B  -  A
)  /  2 ) )  e.  CC )
1917, 18mulcld 7793 . . . 4  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( ( sin `  ( B  -  ( ( B  -  A )  /  2 ) ) )  x.  ( sin `  ( ( B  -  A )  /  2
) ) )  e.  CC )
2016, 19negsubd 8086 . . 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
) ) ) ) )
2110rehalfcld 8973 . . . . . . . 8  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( ( B  -  A )  /  2
)  e.  RR )
226, 21resubcld 8150 . . . . . . 7  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( B  -  (
( B  -  A
)  /  2 ) )  e.  RR )
2322resincld 11437 . . . . . 6  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( sin `  ( B  -  ( ( B  -  A )  /  2 ) ) )  e.  RR )
2421resincld 11437 . . . . . 6  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( sin `  (
( B  -  A
)  /  2 ) )  e.  RR )
2523, 24remulcld 7803 . . . . 5  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( ( sin `  ( B  -  ( ( B  -  A )  /  2 ) ) )  x.  ( sin `  ( ( B  -  A )  /  2
) ) )  e.  RR )
2625renegcld 8149 . . . 4  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  ->  -u ( ( sin `  ( B  -  ( ( B  -  A )  /  2 ) ) )  x.  ( sin `  ( ( B  -  A )  /  2
) ) )  e.  RR )
2722recoscld 11438 . . . . 5  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( cos `  ( B  -  ( ( B  -  A )  /  2 ) ) )  e.  RR )
2821recoscld 11438 . . . . 5  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( cos `  (
( B  -  A
)  /  2 ) )  e.  RR )
2927, 28remulcld 7803 . . . 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 7774 . . . . 5  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
0  e.  RR )
311a1i 9 . . . . . . . . . . 11  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
1  e.  RR )
3231rehalfcld 8973 . . . . . . . . . 10  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( 1  /  2
)  e.  RR )
33 simp3 983 . . . . . . . . . . . . 13  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  ->  A  <  B )
349, 6posdifd 8301 . . . . . . . . . . . . 13  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( A  <  B  <->  0  <  ( B  -  A ) ) )
3533, 34mpbid 146 . . . . . . . . . . . 12  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
0  <  ( B  -  A ) )
36 halfpos2 8957 . . . . . . . . . . . . 13  |-  ( ( B  -  A )  e.  RR  ->  (
0  <  ( B  -  A )  <->  0  <  ( ( B  -  A
)  /  2 ) ) )
3710, 36syl 14 . . . . . . . . . . . 12  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( 0  <  ( B  -  A )  <->  0  <  ( ( B  -  A )  / 
2 ) ) )
3835, 37mpbid 146 . . . . . . . . . . 11  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
0  <  ( ( B  -  A )  /  2 ) )
39 2rp 9453 . . . . . . . . . . . . 13  |-  2  e.  RR+
4039a1i 9 . . . . . . . . . . . 12  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
2  e.  RR+ )
412a1i 9 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
2  e.  RR )
421rexri 7830 . . . . . . . . . . . . . . . 16  |-  1  e.  RR*
432rexri 7830 . . . . . . . . . . . . . . . 16  |-  2  e.  RR*
44 iccleub 9721 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR*  /\  2  e.  RR*  /\  B  e.  ( 1 [,] 2
) )  ->  B  <_  2 )
4542, 43, 44mp3an12 1305 . . . . . . . . . . . . . . 15  |-  ( B  e.  ( 1 [,] 2 )  ->  B  <_  2 )
46453ad2ant2 1003 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  ->  B  <_  2 )
47 iccgelb 9722 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR*  /\  2  e.  RR*  /\  A  e.  ( 1 [,] 2
) )  ->  1  <_  A )
4842, 43, 47mp3an12 1305 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( 1 [,] 2 )  ->  1  <_  A )
49483ad2ant1 1002 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
1  <_  A )
506, 31, 41, 9, 46, 49le2subd 8333 . . . . . . . . . . . . 13  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( B  -  A
)  <_  ( 2  -  1 ) )
51 2m1e1 8845 . . . . . . . . . . . . 13  |-  ( 2  -  1 )  =  1
5250, 51breqtrdi 3969 . . . . . . . . . . . 12  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( B  -  A
)  <_  1 )
5310, 31, 40, 52lediv1dd 9549 . . . . . . . . . . 11  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( ( B  -  A )  /  2
)  <_  ( 1  /  2 ) )
5430, 21, 32, 38, 53ltletrd 8192 . . . . . . . . . 10  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
0  <  ( 1  /  2 ) )
55 1mhlfehlf 8945 . . . . . . . . . . 11  |-  ( 1  -  ( 1  / 
2 ) )  =  ( 1  /  2
)
56 iccgelb 9722 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  RR*  /\  2  e.  RR*  /\  B  e.  ( 1 [,] 2
) )  ->  1  <_  B )
5742, 43, 56mp3an12 1305 . . . . . . . . . . . . 13  |-  ( B  e.  ( 1 [,] 2 )  ->  1  <_  B )
58573ad2ant2 1003 . . . . . . . . . . . 12  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
1  <_  B )
5931, 21, 6, 32, 58, 53le2subd 8333 . . . . . . . . . . 11  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( 1  -  (
1  /  2 ) )  <_  ( B  -  ( ( B  -  A )  / 
2 ) ) )
6055, 59eqbrtrrid 3964 . . . . . . . . . 10  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( 1  /  2
)  <_  ( B  -  ( ( B  -  A )  / 
2 ) ) )
6130, 32, 22, 54, 60ltletrd 8192 . . . . . . . . 9  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
0  <  ( B  -  ( ( B  -  A )  / 
2 ) ) )
6230, 21, 38ltled 7888 . . . . . . . . . . 11  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
0  <_  ( ( B  -  A )  /  2 ) )
636, 30, 41, 21, 46, 62le2subd 8333 . . . . . . . . . 10  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( B  -  (
( B  -  A
)  /  2 ) )  <_  ( 2  -  0 ) )
64 2cn 8798 . . . . . . . . . . 11  |-  2  e.  CC
6564subid1i 8041 . . . . . . . . . 10  |-  ( 2  -  0 )  =  2
6663, 65breqtrdi 3969 . . . . . . . . 9  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( B  -  (
( B  -  A
)  /  2 ) )  <_  2 )
67 0xr 7819 . . . . . . . . . 10  |-  0  e.  RR*
68 elioc2 9726 . . . . . . . . . 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 ) ) )
6967, 2, 68mp2an 422 . . . . . . . . 9  |-  ( ( B  -  ( ( B  -  A )  /  2 ) )  e.  ( 0 (,] 2 )  <->  ( ( B  -  ( ( B  -  A )  /  2 ) )  e.  RR  /\  0  <  ( B  -  (
( B  -  A
)  /  2 ) )  /\  ( B  -  ( ( B  -  A )  / 
2 ) )  <_ 
2 ) )
7022, 61, 66, 69syl3anbrc 1165 . . . . . . . 8  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( B  -  (
( B  -  A
)  /  2 ) )  e.  ( 0 (,] 2 ) )
71 sin02gt0 11477 . . . . . . . 8  |-  ( ( B  -  ( ( B  -  A )  /  2 ) )  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  ( B  -  ( ( B  -  A )  /  2 ) ) ) )
7270, 71syl 14 . . . . . . 7  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
0  <  ( sin `  ( B  -  (
( B  -  A
)  /  2 ) ) ) )
73 halfre 8940 . . . . . . . . . . . 12  |-  ( 1  /  2 )  e.  RR
74 halflt1 8944 . . . . . . . . . . . . 13  |-  ( 1  /  2 )  <  1
75 1lt2 8896 . . . . . . . . . . . . 13  |-  1  <  2
7673, 1, 2lttri 7875 . . . . . . . . . . . . 13  |-  ( ( ( 1  /  2
)  <  1  /\  1  <  2 )  -> 
( 1  /  2
)  <  2 )
7774, 75, 76mp2an 422 . . . . . . . . . . . 12  |-  ( 1  /  2 )  <  2
7873, 2, 77ltleii 7873 . . . . . . . . . . 11  |-  ( 1  /  2 )  <_ 
2
7978a1i 9 . . . . . . . . . 10  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( 1  /  2
)  <_  2 )
8021, 32, 41, 53, 79letrd 7893 . . . . . . . . 9  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( ( B  -  A )  /  2
)  <_  2 )
81 elioc2 9726 . . . . . . . . . 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 ) ) )
8267, 2, 81mp2an 422 . . . . . . . . 9  |-  ( ( ( B  -  A
)  /  2 )  e.  ( 0 (,] 2 )  <->  ( (
( B  -  A
)  /  2 )  e.  RR  /\  0  <  ( ( B  -  A )  /  2
)  /\  ( ( B  -  A )  /  2 )  <_ 
2 ) )
8321, 38, 80, 82syl3anbrc 1165 . . . . . . . 8  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( ( B  -  A )  /  2
)  e.  ( 0 (,] 2 ) )
84 sin02gt0 11477 . . . . . . . 8  |-  ( ( ( B  -  A
)  /  2 )  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  (
( B  -  A
)  /  2 ) ) )
8583, 84syl 14 . . . . . . 7  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
0  <  ( sin `  ( ( B  -  A )  /  2
) ) )
8623, 24, 72, 85mulgt0d 7892 . . . . . 6  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
0  <  ( ( sin `  ( B  -  ( ( B  -  A )  /  2
) ) )  x.  ( sin `  (
( B  -  A
)  /  2 ) ) ) )
8725lt0neg2d 8285 . . . . . 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
) )
8886, 87mpbid 146 . . . . 5  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  ->  -u ( ( sin `  ( B  -  ( ( B  -  A )  /  2 ) ) )  x.  ( sin `  ( ( B  -  A )  /  2
) ) )  <  0 )
8926, 30, 25, 88, 86lttrd 7895 . . . 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
) ) ) )
9026, 25, 29, 89ltadd2dd 8191 . . 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
) ) ) ) )
9120, 90eqbrtrrd 3952 . 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
) ) ) ) )
927, 12npcand 8084 . . . 4  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( ( B  -  ( ( B  -  A )  /  2
) )  +  ( ( B  -  A
)  /  2 ) )  =  B )
9392fveq2d 5425 . . 3  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( cos `  (
( B  -  (
( B  -  A
)  /  2 ) )  +  ( ( B  -  A )  /  2 ) ) )  =  ( cos `  B ) )
94 cosadd 11451 . . . 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
) ) ) ) )
9513, 12, 94syl2anc 408 . . 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
) ) ) ) )
9693, 95eqtr3d 2174 . 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
) ) ) ) )
977, 12, 12subsub4d 8111 . . . . 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
) ) ) )
98112halvesd 8972 . . . . . 6  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( ( ( B  -  A )  / 
2 )  +  ( ( B  -  A
)  /  2 ) )  =  ( B  -  A ) )
9998oveq2d 5790 . . . . 5  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( B  -  (
( ( B  -  A )  /  2
)  +  ( ( B  -  A )  /  2 ) ) )  =  ( B  -  ( B  -  A ) ) )
1009recnd 7801 . . . . . 6  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  ->  A  e.  CC )
1017, 100nncand 8085 . . . . 5  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( B  -  ( B  -  A )
)  =  A )
10297, 99, 1013eqtrd 2176 . . . 4  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( ( B  -  ( ( B  -  A )  /  2
) )  -  (
( B  -  A
)  /  2 ) )  =  A )
103102fveq2d 5425 . . 3  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( cos `  (
( B  -  (
( B  -  A
)  /  2 ) )  -  ( ( B  -  A )  /  2 ) ) )  =  ( cos `  A ) )
104 cossub 11455 . . . 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
) ) ) ) )
10513, 12, 104syl2anc 408 . . 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
) ) ) ) )
106103, 105eqtr3d 2174 . 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
) ) ) ) )
10791, 96, 1063brtr4d 3960 1  |-  ( ( A  e.  ( 1 [,] 2 )  /\  B  e.  ( 1 [,] 2 )  /\  A  <  B )  -> 
( cos `  B
)  <  ( cos `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480    C_ wss 3071   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7625   RRcr 7626   0cc0 7627   1c1 7628    + caddc 7630    x. cmul 7632   RR*cxr 7806    < clt 7807    <_ cle 7808    - cmin 7940   -ucneg 7941    / cdiv 8439   2c2 8778   RR+crp 9448   (,]cioc 9679   [,]cicc 9681   sincsin 11357   cosccos 11358
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-disj 3907  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-sup 6871  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-5 8789  df-6 8790  df-7 8791  df-8 8792  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-ioc 9683  df-ico 9684  df-icc 9685  df-fz 9798  df-fzo 9927  df-seqfrec 10226  df-exp 10300  df-fac 10479  df-bc 10501  df-ihash 10529  df-shft 10594  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-clim 11055  df-sumdc 11130  df-ef 11361  df-sin 11363  df-cos 11364
This theorem is referenced by:  cosz12  12871
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