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Theorem sincos6thpi 13557
Description: The sine and cosine of  pi  /  6. (Contributed by Paul Chapman, 25-Jan-2008.) (Revised by Wolf Lammen, 24-Sep-2020.)
Assertion
Ref Expression
sincos6thpi  |-  ( ( sin `  ( pi 
/  6 ) )  =  ( 1  / 
2 )  /\  ( cos `  ( pi  / 
6 ) )  =  ( ( sqr `  3
)  /  2 ) )

Proof of Theorem sincos6thpi
StepHypRef Expression
1 2cn 8949 . . . . 5  |-  2  e.  CC
21a1i 9 . . . 4  |-  ( T. 
->  2  e.  CC )
3 pire 13501 . . . . . . . 8  |-  pi  e.  RR
4 6re 8959 . . . . . . . 8  |-  6  e.  RR
5 6pos 8979 . . . . . . . . 9  |-  0  <  6
64, 5gt0ap0ii 8547 . . . . . . . 8  |-  6 #  0
73, 4, 6redivclapi 8696 . . . . . . 7  |-  ( pi 
/  6 )  e.  RR
87recni 7932 . . . . . 6  |-  ( pi 
/  6 )  e.  CC
9 sincl 11669 . . . . . 6  |-  ( ( pi  /  6 )  e.  CC  ->  ( sin `  ( pi  / 
6 ) )  e.  CC )
108, 9ax-mp 5 . . . . 5  |-  ( sin `  ( pi  /  6
) )  e.  CC
1110a1i 9 . . . 4  |-  ( T. 
->  ( sin `  (
pi  /  6 ) )  e.  CC )
12 2ap0 8971 . . . . 5  |-  2 #  0
1312a1i 9 . . . 4  |-  ( T. 
->  2 #  0 )
14 recoscl 11684 . . . . . . . . . . . 12  |-  ( ( pi  /  6 )  e.  RR  ->  ( cos `  ( pi  / 
6 ) )  e.  RR )
157, 14ax-mp 5 . . . . . . . . . . 11  |-  ( cos `  ( pi  /  6
) )  e.  RR
1615recni 7932 . . . . . . . . . 10  |-  ( cos `  ( pi  /  6
) )  e.  CC
171, 10, 16mulassi 7929 . . . . . . . . 9  |-  ( ( 2  x.  ( sin `  ( pi  /  6
) ) )  x.  ( cos `  (
pi  /  6 ) ) )  =  ( 2  x.  ( ( sin `  ( pi 
/  6 ) )  x.  ( cos `  (
pi  /  6 ) ) ) )
18 sin2t 11712 . . . . . . . . . 10  |-  ( ( pi  /  6 )  e.  CC  ->  ( sin `  ( 2  x.  ( pi  /  6
) ) )  =  ( 2  x.  (
( sin `  (
pi  /  6 ) )  x.  ( cos `  ( pi  /  6
) ) ) ) )
198, 18ax-mp 5 . . . . . . . . 9  |-  ( sin `  ( 2  x.  (
pi  /  6 ) ) )  =  ( 2  x.  ( ( sin `  ( pi 
/  6 ) )  x.  ( cos `  (
pi  /  6 ) ) ) )
2017, 19eqtr4i 2194 . . . . . . . 8  |-  ( ( 2  x.  ( sin `  ( pi  /  6
) ) )  x.  ( cos `  (
pi  /  6 ) ) )  =  ( sin `  ( 2  x.  ( pi  / 
6 ) ) )
21 3cn 8953 . . . . . . . . . . . 12  |-  3  e.  CC
22 3ap0 8974 . . . . . . . . . . . 12  |-  3 #  0
231, 21, 22divclapi 8671 . . . . . . . . . . 11  |-  ( 2  /  3 )  e.  CC
2421, 22recclapi 8659 . . . . . . . . . . 11  |-  ( 1  /  3 )  e.  CC
25 df-3 8938 . . . . . . . . . . . . 13  |-  3  =  ( 2  +  1 )
2625oveq1i 5863 . . . . . . . . . . . 12  |-  ( 3  /  3 )  =  ( ( 2  +  1 )  /  3
)
2721, 22dividapi 8662 . . . . . . . . . . . 12  |-  ( 3  /  3 )  =  1
28 ax-1cn 7867 . . . . . . . . . . . . 13  |-  1  e.  CC
291, 28, 21, 22divdirapi 8686 . . . . . . . . . . . 12  |-  ( ( 2  +  1 )  /  3 )  =  ( ( 2  / 
3 )  +  ( 1  /  3 ) )
3026, 27, 293eqtr3ri 2200 . . . . . . . . . . 11  |-  ( ( 2  /  3 )  +  ( 1  / 
3 ) )  =  1
31 sincosq1eq 13554 . . . . . . . . . . 11  |-  ( ( ( 2  /  3
)  e.  CC  /\  ( 1  /  3
)  e.  CC  /\  ( ( 2  / 
3 )  +  ( 1  /  3 ) )  =  1 )  ->  ( sin `  (
( 2  /  3
)  x.  ( pi 
/  2 ) ) )  =  ( cos `  ( ( 1  / 
3 )  x.  (
pi  /  2 ) ) ) )
3223, 24, 30, 31mp3an 1332 . . . . . . . . . 10  |-  ( sin `  ( ( 2  / 
3 )  x.  (
pi  /  2 ) ) )  =  ( cos `  ( ( 1  /  3 )  x.  ( pi  / 
2 ) ) )
33 picn 13502 . . . . . . . . . . . . 13  |-  pi  e.  CC
341, 21, 33, 1, 22, 12divmuldivapi 8689 . . . . . . . . . . . 12  |-  ( ( 2  /  3 )  x.  ( pi  / 
2 ) )  =  ( ( 2  x.  pi )  /  (
3  x.  2 ) )
35 3t2e6 9034 . . . . . . . . . . . . 13  |-  ( 3  x.  2 )  =  6
3635oveq2i 5864 . . . . . . . . . . . 12  |-  ( ( 2  x.  pi )  /  ( 3  x.  2 ) )  =  ( ( 2  x.  pi )  /  6
)
37 6cn 8960 . . . . . . . . . . . . 13  |-  6  e.  CC
381, 33, 37, 6divassapi 8685 . . . . . . . . . . . 12  |-  ( ( 2  x.  pi )  /  6 )  =  ( 2  x.  (
pi  /  6 ) )
3934, 36, 383eqtri 2195 . . . . . . . . . . 11  |-  ( ( 2  /  3 )  x.  ( pi  / 
2 ) )  =  ( 2  x.  (
pi  /  6 ) )
4039fveq2i 5499 . . . . . . . . . 10  |-  ( sin `  ( ( 2  / 
3 )  x.  (
pi  /  2 ) ) )  =  ( sin `  ( 2  x.  ( pi  / 
6 ) ) )
4132, 40eqtr3i 2193 . . . . . . . . 9  |-  ( cos `  ( ( 1  / 
3 )  x.  (
pi  /  2 ) ) )  =  ( sin `  ( 2  x.  ( pi  / 
6 ) ) )
4228, 21, 33, 1, 22, 12divmuldivapi 8689 . . . . . . . . . . 11  |-  ( ( 1  /  3 )  x.  ( pi  / 
2 ) )  =  ( ( 1  x.  pi )  /  (
3  x.  2 ) )
4333mulid2i 7923 . . . . . . . . . . . 12  |-  ( 1  x.  pi )  =  pi
4443, 35oveq12i 5865 . . . . . . . . . . 11  |-  ( ( 1  x.  pi )  /  ( 3  x.  2 ) )  =  ( pi  /  6
)
4542, 44eqtri 2191 . . . . . . . . . 10  |-  ( ( 1  /  3 )  x.  ( pi  / 
2 ) )  =  ( pi  /  6
)
4645fveq2i 5499 . . . . . . . . 9  |-  ( cos `  ( ( 1  / 
3 )  x.  (
pi  /  2 ) ) )  =  ( cos `  ( pi 
/  6 ) )
4741, 46eqtr3i 2193 . . . . . . . 8  |-  ( sin `  ( 2  x.  (
pi  /  6 ) ) )  =  ( cos `  ( pi 
/  6 ) )
4820, 47eqtri 2191 . . . . . . 7  |-  ( ( 2  x.  ( sin `  ( pi  /  6
) ) )  x.  ( cos `  (
pi  /  6 ) ) )  =  ( cos `  ( pi 
/  6 ) )
4916mulid2i 7923 . . . . . . 7  |-  ( 1  x.  ( cos `  (
pi  /  6 ) ) )  =  ( cos `  ( pi 
/  6 ) )
5048, 49eqtr4i 2194 . . . . . 6  |-  ( ( 2  x.  ( sin `  ( pi  /  6
) ) )  x.  ( cos `  (
pi  /  6 ) ) )  =  ( 1  x.  ( cos `  ( pi  /  6
) ) )
511, 10mulcli 7925 . . . . . . 7  |-  ( 2  x.  ( sin `  (
pi  /  6 ) ) )  e.  CC
52 pipos 13503 . . . . . . . . . . . . 13  |-  0  <  pi
533, 4, 52, 5divgt0ii 8835 . . . . . . . . . . . 12  |-  0  <  ( pi  /  6
)
54 2lt6 9060 . . . . . . . . . . . . 13  |-  2  <  6
55 2re 8948 . . . . . . . . . . . . . . 15  |-  2  e.  RR
56 2pos 8969 . . . . . . . . . . . . . . 15  |-  0  <  2
5755, 56pm3.2i 270 . . . . . . . . . . . . . 14  |-  ( 2  e.  RR  /\  0  <  2 )
584, 5pm3.2i 270 . . . . . . . . . . . . . 14  |-  ( 6  e.  RR  /\  0  <  6 )
593, 52pm3.2i 270 . . . . . . . . . . . . . 14  |-  ( pi  e.  RR  /\  0  <  pi )
60 ltdiv2 8803 . . . . . . . . . . . . . 14  |-  ( ( ( 2  e.  RR  /\  0  <  2 )  /\  ( 6  e.  RR  /\  0  <  6 )  /\  (
pi  e.  RR  /\  0  <  pi ) )  ->  ( 2  <  6  <->  ( pi  / 
6 )  <  (
pi  /  2 ) ) )
6157, 58, 59, 60mp3an 1332 . . . . . . . . . . . . 13  |-  ( 2  <  6  <->  ( pi  /  6 )  <  (
pi  /  2 ) )
6254, 61mpbi 144 . . . . . . . . . . . 12  |-  ( pi 
/  6 )  < 
( pi  /  2
)
63 0re 7920 . . . . . . . . . . . . 13  |-  0  e.  RR
64 halfpire 13507 . . . . . . . . . . . . 13  |-  ( pi 
/  2 )  e.  RR
65 rexr 7965 . . . . . . . . . . . . . 14  |-  ( 0  e.  RR  ->  0  e.  RR* )
66 rexr 7965 . . . . . . . . . . . . . 14  |-  ( ( pi  /  2 )  e.  RR  ->  (
pi  /  2 )  e.  RR* )
67 elioo2 9878 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR*  /\  (
pi  /  2 )  e.  RR* )  ->  (
( pi  /  6
)  e.  ( 0 (,) ( pi  / 
2 ) )  <->  ( (
pi  /  6 )  e.  RR  /\  0  <  ( pi  /  6
)  /\  ( pi  /  6 )  <  (
pi  /  2 ) ) ) )
6865, 66, 67syl2an 287 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  ( pi  /  2
)  e.  RR )  ->  ( ( pi 
/  6 )  e.  ( 0 (,) (
pi  /  2 ) )  <->  ( ( pi 
/  6 )  e.  RR  /\  0  < 
( pi  /  6
)  /\  ( pi  /  6 )  <  (
pi  /  2 ) ) ) )
6963, 64, 68mp2an 424 . . . . . . . . . . . 12  |-  ( ( pi  /  6 )  e.  ( 0 (,) ( pi  /  2
) )  <->  ( (
pi  /  6 )  e.  RR  /\  0  <  ( pi  /  6
)  /\  ( pi  /  6 )  <  (
pi  /  2 ) ) )
707, 53, 62, 69mpbir3an 1174 . . . . . . . . . . 11  |-  ( pi 
/  6 )  e.  ( 0 (,) (
pi  /  2 ) )
71 sincosq1sgn 13541 . . . . . . . . . . 11  |-  ( ( pi  /  6 )  e.  ( 0 (,) ( pi  /  2
) )  ->  (
0  <  ( sin `  ( pi  /  6
) )  /\  0  <  ( cos `  (
pi  /  6 ) ) ) )
7270, 71ax-mp 5 . . . . . . . . . 10  |-  ( 0  <  ( sin `  (
pi  /  6 ) )  /\  0  < 
( cos `  (
pi  /  6 ) ) )
7372simpri 112 . . . . . . . . 9  |-  0  <  ( cos `  (
pi  /  6 ) )
7415, 73gt0ap0ii 8547 . . . . . . . 8  |-  ( cos `  ( pi  /  6
) ) #  0
7516, 74pm3.2i 270 . . . . . . 7  |-  ( ( cos `  ( pi 
/  6 ) )  e.  CC  /\  ( cos `  ( pi  / 
6 ) ) #  0 )
76 mulcanap2 8584 . . . . . . 7  |-  ( ( ( 2  x.  ( sin `  ( pi  / 
6 ) ) )  e.  CC  /\  1  e.  CC  /\  ( ( cos `  ( pi 
/  6 ) )  e.  CC  /\  ( cos `  ( pi  / 
6 ) ) #  0 ) )  ->  (
( ( 2  x.  ( sin `  (
pi  /  6 ) ) )  x.  ( cos `  ( pi  / 
6 ) ) )  =  ( 1  x.  ( cos `  (
pi  /  6 ) ) )  <->  ( 2  x.  ( sin `  (
pi  /  6 ) ) )  =  1 ) )
7751, 28, 75, 76mp3an 1332 . . . . . 6  |-  ( ( ( 2  x.  ( sin `  ( pi  / 
6 ) ) )  x.  ( cos `  (
pi  /  6 ) ) )  =  ( 1  x.  ( cos `  ( pi  /  6
) ) )  <->  ( 2  x.  ( sin `  (
pi  /  6 ) ) )  =  1 )
7850, 77mpbi 144 . . . . 5  |-  ( 2  x.  ( sin `  (
pi  /  6 ) ) )  =  1
7978a1i 9 . . . 4  |-  ( T. 
->  ( 2  x.  ( sin `  ( pi  / 
6 ) ) )  =  1 )
802, 11, 13, 79mvllmulapd 8759 . . 3  |-  ( T. 
->  ( sin `  (
pi  /  6 ) )  =  ( 1  /  2 ) )
8180mptru 1357 . 2  |-  ( sin `  ( pi  /  6
) )  =  ( 1  /  2 )
82 3re 8952 . . . . . . . 8  |-  3  e.  RR
83 3pos 8972 . . . . . . . 8  |-  0  <  3
8482, 83sqrtpclii 11094 . . . . . . 7  |-  ( sqr `  3 )  e.  RR
8584recni 7932 . . . . . 6  |-  ( sqr `  3 )  e.  CC
8685, 1, 12sqdivapi 10559 . . . . 5  |-  ( ( ( sqr `  3
)  /  2 ) ^ 2 )  =  ( ( ( sqr `  3 ) ^
2 )  /  (
2 ^ 2 ) )
8763, 82, 83ltleii 8022 . . . . . . 7  |-  0  <_  3
8882sqsqrti 11088 . . . . . . 7  |-  ( 0  <_  3  ->  (
( sqr `  3
) ^ 2 )  =  3 )
8987, 88ax-mp 5 . . . . . 6  |-  ( ( sqr `  3 ) ^ 2 )  =  3
90 sq2 10571 . . . . . 6  |-  ( 2 ^ 2 )  =  4
9189, 90oveq12i 5865 . . . . 5  |-  ( ( ( sqr `  3
) ^ 2 )  /  ( 2 ^ 2 ) )  =  ( 3  /  4
)
9286, 91eqtri 2191 . . . 4  |-  ( ( ( sqr `  3
)  /  2 ) ^ 2 )  =  ( 3  /  4
)
9392fveq2i 5499 . . 3  |-  ( sqr `  ( ( ( sqr `  3 )  / 
2 ) ^ 2 ) )  =  ( sqr `  ( 3  /  4 ) )
9482sqrtge0i 11089 . . . . . 6  |-  ( 0  <_  3  ->  0  <_  ( sqr `  3
) )
9587, 94ax-mp 5 . . . . 5  |-  0  <_  ( sqr `  3
)
9684, 55divge0i 8827 . . . . 5  |-  ( ( 0  <_  ( sqr `  3 )  /\  0  <  2 )  ->  0  <_  ( ( sqr `  3
)  /  2 ) )
9795, 56, 96mp2an 424 . . . 4  |-  0  <_  ( ( sqr `  3
)  /  2 )
9884, 55, 12redivclapi 8696 . . . . 5  |-  ( ( sqr `  3 )  /  2 )  e.  RR
9998sqrtsqi 11087 . . . 4  |-  ( 0  <_  ( ( sqr `  3 )  / 
2 )  ->  ( sqr `  ( ( ( sqr `  3 )  /  2 ) ^
2 ) )  =  ( ( sqr `  3
)  /  2 ) )
10097, 99ax-mp 5 . . 3  |-  ( sqr `  ( ( ( sqr `  3 )  / 
2 ) ^ 2 ) )  =  ( ( sqr `  3
)  /  2 )
101 4cn 8956 . . . . . . . 8  |-  4  e.  CC
102 4ap0 8977 . . . . . . . 8  |-  4 #  0
103101, 102dividapi 8662 . . . . . . 7  |-  ( 4  /  4 )  =  1
104103oveq1i 5863 . . . . . 6  |-  ( ( 4  /  4 )  -  ( 1  / 
4 ) )  =  ( 1  -  (
1  /  4 ) )
105101, 102pm3.2i 270 . . . . . . . 8  |-  ( 4  e.  CC  /\  4 #  0 )
106 divsubdirap 8625 . . . . . . . 8  |-  ( ( 4  e.  CC  /\  1  e.  CC  /\  (
4  e.  CC  /\  4 #  0 ) )  -> 
( ( 4  -  1 )  /  4
)  =  ( ( 4  /  4 )  -  ( 1  / 
4 ) ) )
107101, 28, 105, 106mp3an 1332 . . . . . . 7  |-  ( ( 4  -  1 )  /  4 )  =  ( ( 4  / 
4 )  -  (
1  /  4 ) )
108 4m1e3 8999 . . . . . . . 8  |-  ( 4  -  1 )  =  3
109108oveq1i 5863 . . . . . . 7  |-  ( ( 4  -  1 )  /  4 )  =  ( 3  /  4
)
110107, 109eqtr3i 2193 . . . . . 6  |-  ( ( 4  /  4 )  -  ( 1  / 
4 ) )  =  ( 3  /  4
)
111101, 102recclapi 8659 . . . . . . 7  |-  ( 1  /  4 )  e.  CC
11216sqcli 10556 . . . . . . 7  |-  ( ( cos `  ( pi 
/  6 ) ) ^ 2 )  e.  CC
11381oveq1i 5863 . . . . . . . . . 10  |-  ( ( sin `  ( pi 
/  6 ) ) ^ 2 )  =  ( ( 1  / 
2 ) ^ 2 )
114 2z 9240 . . . . . . . . . . 11  |-  2  e.  ZZ
115 exprecap 10517 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  2 #  0  /\  2  e.  ZZ )  ->  (
( 1  /  2
) ^ 2 )  =  ( 1  / 
( 2 ^ 2 ) ) )
1161, 12, 114, 115mp3an 1332 . . . . . . . . . 10  |-  ( ( 1  /  2 ) ^ 2 )  =  ( 1  /  (
2 ^ 2 ) )
11790oveq2i 5864 . . . . . . . . . 10  |-  ( 1  /  ( 2 ^ 2 ) )  =  ( 1  /  4
)
118113, 116, 1173eqtri 2195 . . . . . . . . 9  |-  ( ( sin `  ( pi 
/  6 ) ) ^ 2 )  =  ( 1  /  4
)
119118oveq1i 5863 . . . . . . . 8  |-  ( ( ( sin `  (
pi  /  6 ) ) ^ 2 )  +  ( ( cos `  ( pi  /  6
) ) ^ 2 ) )  =  ( ( 1  /  4
)  +  ( ( cos `  ( pi 
/  6 ) ) ^ 2 ) )
120 sincossq 11711 . . . . . . . . 9  |-  ( ( pi  /  6 )  e.  CC  ->  (
( ( sin `  (
pi  /  6 ) ) ^ 2 )  +  ( ( cos `  ( pi  /  6
) ) ^ 2 ) )  =  1 )
1218, 120ax-mp 5 . . . . . . . 8  |-  ( ( ( sin `  (
pi  /  6 ) ) ^ 2 )  +  ( ( cos `  ( pi  /  6
) ) ^ 2 ) )  =  1
122119, 121eqtr3i 2193 . . . . . . 7  |-  ( ( 1  /  4 )  +  ( ( cos `  ( pi  /  6
) ) ^ 2 ) )  =  1
12328, 111, 112, 122subaddrii 8208 . . . . . 6  |-  ( 1  -  ( 1  / 
4 ) )  =  ( ( cos `  (
pi  /  6 ) ) ^ 2 )
124104, 110, 1233eqtr3ri 2200 . . . . 5  |-  ( ( cos `  ( pi 
/  6 ) ) ^ 2 )  =  ( 3  /  4
)
125124fveq2i 5499 . . . 4  |-  ( sqr `  ( ( cos `  (
pi  /  6 ) ) ^ 2 ) )  =  ( sqr `  ( 3  /  4
) )
12663, 15, 73ltleii 8022 . . . . 5  |-  0  <_  ( cos `  (
pi  /  6 ) )
12715sqrtsqi 11087 . . . . 5  |-  ( 0  <_  ( cos `  (
pi  /  6 ) )  ->  ( sqr `  ( ( cos `  (
pi  /  6 ) ) ^ 2 ) )  =  ( cos `  ( pi  /  6
) ) )
128126, 127ax-mp 5 . . . 4  |-  ( sqr `  ( ( cos `  (
pi  /  6 ) ) ^ 2 ) )  =  ( cos `  ( pi  /  6
) )
129125, 128eqtr3i 2193 . . 3  |-  ( sqr `  ( 3  /  4
) )  =  ( cos `  ( pi 
/  6 ) )
13093, 100, 1293eqtr3ri 2200 . 2  |-  ( cos `  ( pi  /  6
) )  =  ( ( sqr `  3
)  /  2 )
13181, 130pm3.2i 270 1  |-  ( ( sin `  ( pi 
/  6 ) )  =  ( 1  / 
2 )  /\  ( cos `  ( pi  / 
6 ) )  =  ( ( sqr `  3
)  /  2 ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    /\ w3a 973    = wceq 1348   T. wtru 1349    e. wcel 2141   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   # cap 8500    / cdiv 8589   2c2 8929   3c3 8930   4c4 8931   6c6 8933   ZZcz 9212   (,)cioo 9845   ^cexp 10475   sqrcsqrt 10960   sincsin 11607   cosccos 11608   picpi 11610
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  ax-pre-suploc 7895  ax-addf 7896  ax-mulf 7897
This theorem depends on definitions:  df-bi 116  df-stab 826  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-of 6061  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-map 6628  df-pm 6629  df-en 6719  df-dom 6720  df-fin 6721  df-sup 6961  df-inf 6962  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-9 8944  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-xneg 9729  df-xadd 9730  df-ioo 9849  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  df-pi 11616  df-rest 12581  df-topgen 12600  df-psmet 12781  df-xmet 12782  df-met 12783  df-bl 12784  df-mopn 12785  df-top 12790  df-topon 12803  df-bases 12835  df-ntr 12890  df-cn 12982  df-cnp 12983  df-tx 13047  df-cncf 13352  df-limced 13419  df-dvap 13420
This theorem is referenced by:  sincos3rdpi  13558  pigt3  13559
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