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Theorem sincos6thpi 13930
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 8979 . . . . 5  |-  2  e.  CC
21a1i 9 . . . 4  |-  ( T. 
->  2  e.  CC )
3 pire 13874 . . . . . . . 8  |-  pi  e.  RR
4 6re 8989 . . . . . . . 8  |-  6  e.  RR
5 6pos 9009 . . . . . . . . 9  |-  0  <  6
64, 5gt0ap0ii 8575 . . . . . . . 8  |-  6 #  0
73, 4, 6redivclapi 8725 . . . . . . 7  |-  ( pi 
/  6 )  e.  RR
87recni 7960 . . . . . 6  |-  ( pi 
/  6 )  e.  CC
9 sincl 11698 . . . . . 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 9001 . . . . 5  |-  2 #  0
1312a1i 9 . . . 4  |-  ( T. 
->  2 #  0 )
14 recoscl 11713 . . . . . . . . . . . 12  |-  ( ( pi  /  6 )  e.  RR  ->  ( cos `  ( pi  / 
6 ) )  e.  RR )
157, 14ax-mp 5 . . . . . . . . . . 11  |-  ( cos `  ( pi  /  6
) )  e.  RR
1615recni 7960 . . . . . . . . . 10  |-  ( cos `  ( pi  /  6
) )  e.  CC
171, 10, 16mulassi 7957 . . . . . . . . 9  |-  ( ( 2  x.  ( sin `  ( pi  /  6
) ) )  x.  ( cos `  (
pi  /  6 ) ) )  =  ( 2  x.  ( ( sin `  ( pi 
/  6 ) )  x.  ( cos `  (
pi  /  6 ) ) ) )
18 sin2t 11741 . . . . . . . . . 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 2201 . . . . . . . 8  |-  ( ( 2  x.  ( sin `  ( pi  /  6
) ) )  x.  ( cos `  (
pi  /  6 ) ) )  =  ( sin `  ( 2  x.  ( pi  / 
6 ) ) )
21 3cn 8983 . . . . . . . . . . . 12  |-  3  e.  CC
22 3ap0 9004 . . . . . . . . . . . 12  |-  3 #  0
231, 21, 22divclapi 8700 . . . . . . . . . . 11  |-  ( 2  /  3 )  e.  CC
2421, 22recclapi 8688 . . . . . . . . . . 11  |-  ( 1  /  3 )  e.  CC
25 df-3 8968 . . . . . . . . . . . . 13  |-  3  =  ( 2  +  1 )
2625oveq1i 5879 . . . . . . . . . . . 12  |-  ( 3  /  3 )  =  ( ( 2  +  1 )  /  3
)
2721, 22dividapi 8691 . . . . . . . . . . . 12  |-  ( 3  /  3 )  =  1
28 ax-1cn 7895 . . . . . . . . . . . . 13  |-  1  e.  CC
291, 28, 21, 22divdirapi 8715 . . . . . . . . . . . 12  |-  ( ( 2  +  1 )  /  3 )  =  ( ( 2  / 
3 )  +  ( 1  /  3 ) )
3026, 27, 293eqtr3ri 2207 . . . . . . . . . . 11  |-  ( ( 2  /  3 )  +  ( 1  / 
3 ) )  =  1
31 sincosq1eq 13927 . . . . . . . . . . 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 1337 . . . . . . . . . 10  |-  ( sin `  ( ( 2  / 
3 )  x.  (
pi  /  2 ) ) )  =  ( cos `  ( ( 1  /  3 )  x.  ( pi  / 
2 ) ) )
33 picn 13875 . . . . . . . . . . . . 13  |-  pi  e.  CC
341, 21, 33, 1, 22, 12divmuldivapi 8718 . . . . . . . . . . . 12  |-  ( ( 2  /  3 )  x.  ( pi  / 
2 ) )  =  ( ( 2  x.  pi )  /  (
3  x.  2 ) )
35 3t2e6 9064 . . . . . . . . . . . . 13  |-  ( 3  x.  2 )  =  6
3635oveq2i 5880 . . . . . . . . . . . 12  |-  ( ( 2  x.  pi )  /  ( 3  x.  2 ) )  =  ( ( 2  x.  pi )  /  6
)
37 6cn 8990 . . . . . . . . . . . . 13  |-  6  e.  CC
381, 33, 37, 6divassapi 8714 . . . . . . . . . . . 12  |-  ( ( 2  x.  pi )  /  6 )  =  ( 2  x.  (
pi  /  6 ) )
3934, 36, 383eqtri 2202 . . . . . . . . . . 11  |-  ( ( 2  /  3 )  x.  ( pi  / 
2 ) )  =  ( 2  x.  (
pi  /  6 ) )
4039fveq2i 5514 . . . . . . . . . 10  |-  ( sin `  ( ( 2  / 
3 )  x.  (
pi  /  2 ) ) )  =  ( sin `  ( 2  x.  ( pi  / 
6 ) ) )
4132, 40eqtr3i 2200 . . . . . . . . 9  |-  ( cos `  ( ( 1  / 
3 )  x.  (
pi  /  2 ) ) )  =  ( sin `  ( 2  x.  ( pi  / 
6 ) ) )
4228, 21, 33, 1, 22, 12divmuldivapi 8718 . . . . . . . . . . 11  |-  ( ( 1  /  3 )  x.  ( pi  / 
2 ) )  =  ( ( 1  x.  pi )  /  (
3  x.  2 ) )
4333mulid2i 7951 . . . . . . . . . . . 12  |-  ( 1  x.  pi )  =  pi
4443, 35oveq12i 5881 . . . . . . . . . . 11  |-  ( ( 1  x.  pi )  /  ( 3  x.  2 ) )  =  ( pi  /  6
)
4542, 44eqtri 2198 . . . . . . . . . 10  |-  ( ( 1  /  3 )  x.  ( pi  / 
2 ) )  =  ( pi  /  6
)
4645fveq2i 5514 . . . . . . . . 9  |-  ( cos `  ( ( 1  / 
3 )  x.  (
pi  /  2 ) ) )  =  ( cos `  ( pi 
/  6 ) )
4741, 46eqtr3i 2200 . . . . . . . 8  |-  ( sin `  ( 2  x.  (
pi  /  6 ) ) )  =  ( cos `  ( pi 
/  6 ) )
4820, 47eqtri 2198 . . . . . . 7  |-  ( ( 2  x.  ( sin `  ( pi  /  6
) ) )  x.  ( cos `  (
pi  /  6 ) ) )  =  ( cos `  ( pi 
/  6 ) )
4916mulid2i 7951 . . . . . . 7  |-  ( 1  x.  ( cos `  (
pi  /  6 ) ) )  =  ( cos `  ( pi 
/  6 ) )
5048, 49eqtr4i 2201 . . . . . 6  |-  ( ( 2  x.  ( sin `  ( pi  /  6
) ) )  x.  ( cos `  (
pi  /  6 ) ) )  =  ( 1  x.  ( cos `  ( pi  /  6
) ) )
511, 10mulcli 7953 . . . . . . 7  |-  ( 2  x.  ( sin `  (
pi  /  6 ) ) )  e.  CC
52 pipos 13876 . . . . . . . . . . . . 13  |-  0  <  pi
533, 4, 52, 5divgt0ii 8865 . . . . . . . . . . . 12  |-  0  <  ( pi  /  6
)
54 2lt6 9090 . . . . . . . . . . . . 13  |-  2  <  6
55 2re 8978 . . . . . . . . . . . . . . 15  |-  2  e.  RR
56 2pos 8999 . . . . . . . . . . . . . . 15  |-  0  <  2
5755, 56pm3.2i 272 . . . . . . . . . . . . . 14  |-  ( 2  e.  RR  /\  0  <  2 )
584, 5pm3.2i 272 . . . . . . . . . . . . . 14  |-  ( 6  e.  RR  /\  0  <  6 )
593, 52pm3.2i 272 . . . . . . . . . . . . . 14  |-  ( pi  e.  RR  /\  0  <  pi )
60 ltdiv2 8833 . . . . . . . . . . . . . 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 1337 . . . . . . . . . . . . 13  |-  ( 2  <  6  <->  ( pi  /  6 )  <  (
pi  /  2 ) )
6254, 61mpbi 145 . . . . . . . . . . . 12  |-  ( pi 
/  6 )  < 
( pi  /  2
)
63 0re 7948 . . . . . . . . . . . . 13  |-  0  e.  RR
64 halfpire 13880 . . . . . . . . . . . . 13  |-  ( pi 
/  2 )  e.  RR
65 rexr 7993 . . . . . . . . . . . . . 14  |-  ( 0  e.  RR  ->  0  e.  RR* )
66 rexr 7993 . . . . . . . . . . . . . 14  |-  ( ( pi  /  2 )  e.  RR  ->  (
pi  /  2 )  e.  RR* )
67 elioo2 9908 . . . . . . . . . . . . . 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 289 . . . . . . . . . . . . 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 426 . . . . . . . . . . . 12  |-  ( ( pi  /  6 )  e.  ( 0 (,) ( pi  /  2
) )  <->  ( (
pi  /  6 )  e.  RR  /\  0  <  ( pi  /  6
)  /\  ( pi  /  6 )  <  (
pi  /  2 ) ) )
707, 53, 62, 69mpbir3an 1179 . . . . . . . . . . 11  |-  ( pi 
/  6 )  e.  ( 0 (,) (
pi  /  2 ) )
71 sincosq1sgn 13914 . . . . . . . . . . 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 113 . . . . . . . . 9  |-  0  <  ( cos `  (
pi  /  6 ) )
7415, 73gt0ap0ii 8575 . . . . . . . 8  |-  ( cos `  ( pi  /  6
) ) #  0
7516, 74pm3.2i 272 . . . . . . 7  |-  ( ( cos `  ( pi 
/  6 ) )  e.  CC  /\  ( cos `  ( pi  / 
6 ) ) #  0 )
76 mulcanap2 8612 . . . . . . 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 1337 . . . . . 6  |-  ( ( ( 2  x.  ( sin `  ( pi  / 
6 ) ) )  x.  ( cos `  (
pi  /  6 ) ) )  =  ( 1  x.  ( cos `  ( pi  /  6
) ) )  <->  ( 2  x.  ( sin `  (
pi  /  6 ) ) )  =  1 )
7850, 77mpbi 145 . . . . 5  |-  ( 2  x.  ( sin `  (
pi  /  6 ) ) )  =  1
7978a1i 9 . . . 4  |-  ( T. 
->  ( 2  x.  ( sin `  ( pi  / 
6 ) ) )  =  1 )
802, 11, 13, 79mvllmulapd 8788 . . 3  |-  ( T. 
->  ( sin `  (
pi  /  6 ) )  =  ( 1  /  2 ) )
8180mptru 1362 . 2  |-  ( sin `  ( pi  /  6
) )  =  ( 1  /  2 )
82 3re 8982 . . . . . . . 8  |-  3  e.  RR
83 3pos 9002 . . . . . . . 8  |-  0  <  3
8482, 83sqrtpclii 11123 . . . . . . 7  |-  ( sqr `  3 )  e.  RR
8584recni 7960 . . . . . 6  |-  ( sqr `  3 )  e.  CC
8685, 1, 12sqdivapi 10589 . . . . 5  |-  ( ( ( sqr `  3
)  /  2 ) ^ 2 )  =  ( ( ( sqr `  3 ) ^
2 )  /  (
2 ^ 2 ) )
8763, 82, 83ltleii 8050 . . . . . . 7  |-  0  <_  3
8882sqsqrti 11117 . . . . . . 7  |-  ( 0  <_  3  ->  (
( sqr `  3
) ^ 2 )  =  3 )
8987, 88ax-mp 5 . . . . . 6  |-  ( ( sqr `  3 ) ^ 2 )  =  3
90 sq2 10601 . . . . . 6  |-  ( 2 ^ 2 )  =  4
9189, 90oveq12i 5881 . . . . 5  |-  ( ( ( sqr `  3
) ^ 2 )  /  ( 2 ^ 2 ) )  =  ( 3  /  4
)
9286, 91eqtri 2198 . . . 4  |-  ( ( ( sqr `  3
)  /  2 ) ^ 2 )  =  ( 3  /  4
)
9392fveq2i 5514 . . 3  |-  ( sqr `  ( ( ( sqr `  3 )  / 
2 ) ^ 2 ) )  =  ( sqr `  ( 3  /  4 ) )
9482sqrtge0i 11118 . . . . . 6  |-  ( 0  <_  3  ->  0  <_  ( sqr `  3
) )
9587, 94ax-mp 5 . . . . 5  |-  0  <_  ( sqr `  3
)
9684, 55divge0i 8857 . . . . 5  |-  ( ( 0  <_  ( sqr `  3 )  /\  0  <  2 )  ->  0  <_  ( ( sqr `  3
)  /  2 ) )
9795, 56, 96mp2an 426 . . . 4  |-  0  <_  ( ( sqr `  3
)  /  2 )
9884, 55, 12redivclapi 8725 . . . . 5  |-  ( ( sqr `  3 )  /  2 )  e.  RR
9998sqrtsqi 11116 . . . 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 8986 . . . . . . . 8  |-  4  e.  CC
102 4ap0 9007 . . . . . . . 8  |-  4 #  0
103101, 102dividapi 8691 . . . . . . 7  |-  ( 4  /  4 )  =  1
104103oveq1i 5879 . . . . . 6  |-  ( ( 4  /  4 )  -  ( 1  / 
4 ) )  =  ( 1  -  (
1  /  4 ) )
105101, 102pm3.2i 272 . . . . . . . 8  |-  ( 4  e.  CC  /\  4 #  0 )
106 divsubdirap 8654 . . . . . . . 8  |-  ( ( 4  e.  CC  /\  1  e.  CC  /\  (
4  e.  CC  /\  4 #  0 ) )  -> 
( ( 4  -  1 )  /  4
)  =  ( ( 4  /  4 )  -  ( 1  / 
4 ) ) )
107101, 28, 105, 106mp3an 1337 . . . . . . 7  |-  ( ( 4  -  1 )  /  4 )  =  ( ( 4  / 
4 )  -  (
1  /  4 ) )
108 4m1e3 9029 . . . . . . . 8  |-  ( 4  -  1 )  =  3
109108oveq1i 5879 . . . . . . 7  |-  ( ( 4  -  1 )  /  4 )  =  ( 3  /  4
)
110107, 109eqtr3i 2200 . . . . . 6  |-  ( ( 4  /  4 )  -  ( 1  / 
4 ) )  =  ( 3  /  4
)
111101, 102recclapi 8688 . . . . . . 7  |-  ( 1  /  4 )  e.  CC
11216sqcli 10586 . . . . . . 7  |-  ( ( cos `  ( pi 
/  6 ) ) ^ 2 )  e.  CC
11381oveq1i 5879 . . . . . . . . . 10  |-  ( ( sin `  ( pi 
/  6 ) ) ^ 2 )  =  ( ( 1  / 
2 ) ^ 2 )
114 2z 9270 . . . . . . . . . . 11  |-  2  e.  ZZ
115 exprecap 10547 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  2 #  0  /\  2  e.  ZZ )  ->  (
( 1  /  2
) ^ 2 )  =  ( 1  / 
( 2 ^ 2 ) ) )
1161, 12, 114, 115mp3an 1337 . . . . . . . . . 10  |-  ( ( 1  /  2 ) ^ 2 )  =  ( 1  /  (
2 ^ 2 ) )
11790oveq2i 5880 . . . . . . . . . 10  |-  ( 1  /  ( 2 ^ 2 ) )  =  ( 1  /  4
)
118113, 116, 1173eqtri 2202 . . . . . . . . 9  |-  ( ( sin `  ( pi 
/  6 ) ) ^ 2 )  =  ( 1  /  4
)
119118oveq1i 5879 . . . . . . . 8  |-  ( ( ( sin `  (
pi  /  6 ) ) ^ 2 )  +  ( ( cos `  ( pi  /  6
) ) ^ 2 ) )  =  ( ( 1  /  4
)  +  ( ( cos `  ( pi 
/  6 ) ) ^ 2 ) )
120 sincossq 11740 . . . . . . . . 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 2200 . . . . . . 7  |-  ( ( 1  /  4 )  +  ( ( cos `  ( pi  /  6
) ) ^ 2 ) )  =  1
12328, 111, 112, 122subaddrii 8236 . . . . . 6  |-  ( 1  -  ( 1  / 
4 ) )  =  ( ( cos `  (
pi  /  6 ) ) ^ 2 )
124104, 110, 1233eqtr3ri 2207 . . . . 5  |-  ( ( cos `  ( pi 
/  6 ) ) ^ 2 )  =  ( 3  /  4
)
125124fveq2i 5514 . . . 4  |-  ( sqr `  ( ( cos `  (
pi  /  6 ) ) ^ 2 ) )  =  ( sqr `  ( 3  /  4
) )
12663, 15, 73ltleii 8050 . . . . 5  |-  0  <_  ( cos `  (
pi  /  6 ) )
12715sqrtsqi 11116 . . . . 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 2200 . . 3  |-  ( sqr `  ( 3  /  4
) )  =  ( cos `  ( pi 
/  6 ) )
13093, 100, 1293eqtr3ri 2207 . 2  |-  ( cos `  ( pi  /  6
) )  =  ( ( sqr `  3
)  /  2 )
13181, 130pm3.2i 272 1  |-  ( ( sin `  ( pi 
/  6 ) )  =  ( 1  / 
2 )  /\  ( cos `  ( pi  / 
6 ) )  =  ( ( sqr `  3
)  /  2 ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353   T. wtru 1354    e. wcel 2148   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   # cap 8528    / cdiv 8618   2c2 8959   3c3 8960   4c4 8961   6c6 8963   ZZcz 9242   (,)cioo 9875   ^cexp 10505   sqrcsqrt 10989   sincsin 11636   cosccos 11637   picpi 11639
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  ax-pre-suploc 7923  ax-addf 7924  ax-mulf 7925
This theorem depends on definitions:  df-bi 117  df-stab 831  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-of 6077  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-map 6644  df-pm 6645  df-en 6735  df-dom 6736  df-fin 6737  df-sup 6977  df-inf 6978  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-9 8974  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-xneg 9759  df-xadd 9760  df-ioo 9879  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  df-pi 11645  df-rest 12638  df-topgen 12657  df-psmet 13154  df-xmet 13155  df-met 13156  df-bl 13157  df-mopn 13158  df-top 13163  df-topon 13176  df-bases 13208  df-ntr 13263  df-cn 13355  df-cnp 13356  df-tx 13420  df-cncf 13725  df-limced 13792  df-dvap 13793
This theorem is referenced by:  sincos3rdpi  13931  pigt3  13932
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