ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  coseq0negpitopi Unicode version

Theorem coseq0negpitopi 12927
Description: Location of the zeroes of cosine in  ( -u pi (,] pi ). (Contributed by David Moews, 28-Feb-2017.)
Assertion
Ref Expression
coseq0negpitopi  |-  ( A  e.  ( -u pi (,] pi )  ->  (
( cos `  A
)  =  0  <->  A  e.  { ( pi  / 
2 ) ,  -u ( pi  /  2
) } ) )

Proof of Theorem coseq0negpitopi
StepHypRef Expression
1 simplr 519 . . . . 5  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  ( cos `  A )  =  0 )
2 pire 12877 . . . . . . . . . . . . 13  |-  pi  e.  RR
32renegcli 8031 . . . . . . . . . . . 12  |-  -u pi  e.  RR
43rexri 7830 . . . . . . . . . . 11  |-  -u pi  e.  RR*
5 elioc2 9726 . . . . . . . . . . 11  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR )  ->  ( A  e.  ( -u pi (,] pi )  <->  ( A  e.  RR  /\  -u pi  <  A  /\  A  <_  pi ) ) )
64, 2, 5mp2an 422 . . . . . . . . . 10  |-  ( A  e.  ( -u pi (,] pi )  <->  ( A  e.  RR  /\  -u pi  <  A  /\  A  <_  pi ) )
76simp1bi 996 . . . . . . . . 9  |-  ( A  e.  ( -u pi (,] pi )  ->  A  e.  RR )
87adantr 274 . . . . . . . 8  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  A  e.  RR )
98adantr 274 . . . . . . 7  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  A  e.  RR )
10 halfpire 12883 . . . . . . . . . 10  |-  ( pi 
/  2 )  e.  RR
1110renegcli 8031 . . . . . . . . 9  |-  -u (
pi  /  2 )  e.  RR
1211a1i 9 . . . . . . . 8  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  -u ( pi 
/  2 )  e.  RR )
13 4re 8804 . . . . . . . . . . 11  |-  4  e.  RR
14 4ap0 8826 . . . . . . . . . . 11  |-  4 #  0
152, 13, 14redivclapi 8546 . . . . . . . . . 10  |-  ( pi 
/  4 )  e.  RR
1615renegcli 8031 . . . . . . . . 9  |-  -u (
pi  /  4 )  e.  RR
1716a1i 9 . . . . . . . 8  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  -u ( pi 
/  4 )  e.  RR )
18 2lt4 8900 . . . . . . . . . . 11  |-  2  <  4
19 2re 8797 . . . . . . . . . . . . 13  |-  2  e.  RR
20 2pos 8818 . . . . . . . . . . . . 13  |-  0  <  2
2119, 20pm3.2i 270 . . . . . . . . . . . 12  |-  ( 2  e.  RR  /\  0  <  2 )
22 4pos 8824 . . . . . . . . . . . . 13  |-  0  <  4
2313, 22pm3.2i 270 . . . . . . . . . . . 12  |-  ( 4  e.  RR  /\  0  <  4 )
24 pipos 12879 . . . . . . . . . . . . 13  |-  0  <  pi
252, 24pm3.2i 270 . . . . . . . . . . . 12  |-  ( pi  e.  RR  /\  0  <  pi )
26 ltdiv2 8652 . . . . . . . . . . . 12  |-  ( ( ( 2  e.  RR  /\  0  <  2 )  /\  ( 4  e.  RR  /\  0  <  4 )  /\  (
pi  e.  RR  /\  0  <  pi ) )  ->  ( 2  <  4  <->  ( pi  / 
4 )  <  (
pi  /  2 ) ) )
2721, 23, 25, 26mp3an 1315 . . . . . . . . . . 11  |-  ( 2  <  4  <->  ( pi  /  4 )  <  (
pi  /  2 ) )
2818, 27mpbi 144 . . . . . . . . . 10  |-  ( pi 
/  4 )  < 
( pi  /  2
)
2915, 10ltnegi 8262 . . . . . . . . . 10  |-  ( ( pi  /  4 )  <  ( pi  / 
2 )  <->  -u ( pi 
/  2 )  <  -u ( pi  /  4
) )
3028, 29mpbi 144 . . . . . . . . 9  |-  -u (
pi  /  2 )  <  -u ( pi  / 
4 )
3130a1i 9 . . . . . . . 8  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  -u ( pi 
/  2 )  <  -u ( pi  /  4
) )
32 simpr 109 . . . . . . . 8  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  -u ( pi 
/  4 )  < 
A )
3312, 17, 9, 31, 32lttrd 7895 . . . . . . 7  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  -u ( pi 
/  2 )  < 
A )
342a1i 9 . . . . . . . 8  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  pi  e.  RR )
35 3re 8801 . . . . . . . . . 10  |-  3  e.  RR
3635, 10remulcli 7787 . . . . . . . . 9  |-  ( 3  x.  ( pi  / 
2 ) )  e.  RR
3736a1i 9 . . . . . . . 8  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  ( 3  x.  ( pi  / 
2 ) )  e.  RR )
386simp3bi 998 . . . . . . . . 9  |-  ( A  e.  ( -u pi (,] pi )  ->  A  <_  pi )
3938ad2antrr 479 . . . . . . . 8  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  A  <_  pi )
40 2lt3 8897 . . . . . . . . . . 11  |-  2  <  3
41 3pos 8821 . . . . . . . . . . . . 13  |-  0  <  3
4235, 41pm3.2i 270 . . . . . . . . . . . 12  |-  ( 3  e.  RR  /\  0  <  3 )
43 ltdiv2 8652 . . . . . . . . . . . 12  |-  ( ( ( 2  e.  RR  /\  0  <  2 )  /\  ( 3  e.  RR  /\  0  <  3 )  /\  (
pi  e.  RR  /\  0  <  pi ) )  ->  ( 2  <  3  <->  ( pi  / 
3 )  <  (
pi  /  2 ) ) )
4421, 42, 25, 43mp3an 1315 . . . . . . . . . . 11  |-  ( 2  <  3  <->  ( pi  /  3 )  <  (
pi  /  2 ) )
4540, 44mpbi 144 . . . . . . . . . 10  |-  ( pi 
/  3 )  < 
( pi  /  2
)
46 ltdivmul 8641 . . . . . . . . . . 11  |-  ( ( pi  e.  RR  /\  ( pi  /  2
)  e.  RR  /\  ( 3  e.  RR  /\  0  <  3 ) )  ->  ( (
pi  /  3 )  <  ( pi  / 
2 )  <->  pi  <  ( 3  x.  ( pi 
/  2 ) ) ) )
472, 10, 42, 46mp3an 1315 . . . . . . . . . 10  |-  ( ( pi  /  3 )  <  ( pi  / 
2 )  <->  pi  <  ( 3  x.  ( pi 
/  2 ) ) )
4845, 47mpbi 144 . . . . . . . . 9  |-  pi  <  ( 3  x.  ( pi 
/  2 ) )
4948a1i 9 . . . . . . . 8  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  pi  <  ( 3  x.  ( pi 
/  2 ) ) )
509, 34, 37, 39, 49lelttrd 7894 . . . . . . 7  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  A  <  ( 3  x.  ( pi 
/  2 ) ) )
5111rexri 7830 . . . . . . . 8  |-  -u (
pi  /  2 )  e.  RR*
5236rexri 7830 . . . . . . . 8  |-  ( 3  x.  ( pi  / 
2 ) )  e. 
RR*
53 elioo2 9711 . . . . . . . 8  |-  ( (
-u ( pi  / 
2 )  e.  RR*  /\  ( 3  x.  (
pi  /  2 ) )  e.  RR* )  ->  ( A  e.  (
-u ( pi  / 
2 ) (,) (
3  x.  ( pi 
/  2 ) ) )  <->  ( A  e.  RR  /\  -u (
pi  /  2 )  <  A  /\  A  <  ( 3  x.  (
pi  /  2 ) ) ) ) )
5451, 52, 53mp2an 422 . . . . . . 7  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( 3  x.  ( pi  /  2
) ) )  <->  ( A  e.  RR  /\  -u (
pi  /  2 )  <  A  /\  A  <  ( 3  x.  (
pi  /  2 ) ) ) )
559, 33, 50, 54syl3anbrc 1165 . . . . . 6  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  A  e.  ( -u ( pi  / 
2 ) (,) (
3  x.  ( pi 
/  2 ) ) ) )
56 coseq0q4123 12925 . . . . . 6  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( 3  x.  ( pi  /  2
) ) )  -> 
( ( cos `  A
)  =  0  <->  A  =  ( pi  / 
2 ) ) )
5755, 56syl 14 . . . . 5  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  ( ( cos `  A )  =  0  <->  A  =  (
pi  /  2 ) ) )
581, 57mpbid 146 . . . 4  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  A  =  ( pi  /  2
) )
59 prid1g 3627 . . . . 5  |-  ( ( pi  /  2 )  e.  RR  ->  (
pi  /  2 )  e.  { ( pi 
/  2 ) , 
-u ( pi  / 
2 ) } )
60 eleq1a 2211 . . . . 5  |-  ( ( pi  /  2 )  e.  { ( pi 
/  2 ) , 
-u ( pi  / 
2 ) }  ->  ( A  =  ( pi 
/  2 )  ->  A  e.  { (
pi  /  2 ) ,  -u ( pi  / 
2 ) } ) )
6110, 59, 60mp2b 8 . . . 4  |-  ( A  =  ( pi  / 
2 )  ->  A  e.  { ( pi  / 
2 ) ,  -u ( pi  /  2
) } )
6258, 61syl 14 . . 3  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  -u ( pi  / 
4 )  <  A
)  ->  A  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )
638recnd 7801 . . . . . . 7  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  A  e.  CC )
6463adantr 274 . . . . . 6  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  A  e.  CC )
65 cosneg 11441 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( cos `  -u A )  =  ( cos `  A
) )
6664, 65syl 14 . . . . . . . 8  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  ( cos `  -u A )  =  ( cos `  A ) )
67 simplr 519 . . . . . . . 8  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  ( cos `  A )  =  0 )
6866, 67eqtrd 2172 . . . . . . 7  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  ( cos `  -u A )  =  0 )
698renegcld 8149 . . . . . . . . . 10  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  -u A  e.  RR )
7069adantr 274 . . . . . . . . 9  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  -u A  e.  RR )
71 0re 7773 . . . . . . . . . . 11  |-  0  e.  RR
7271a1i 9 . . . . . . . . . 10  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  0  e.  RR )
73 simpr 109 . . . . . . . . . . 11  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  A  <  0 )
748adantr 274 . . . . . . . . . . . 12  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  A  e.  RR )
7574lt0neg1d 8284 . . . . . . . . . . 11  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  ( A  <  0  <->  0  <  -u A
) )
7673, 75mpbid 146 . . . . . . . . . 10  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  0  <  -u A )
7772, 70, 76ltled 7888 . . . . . . . . 9  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  0  <_  -u A )
782a1i 9 . . . . . . . . . 10  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  pi  e.  RR )
792a1i 9 . . . . . . . . . . . 12  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  pi  e.  RR )
806simp2bi 997 . . . . . . . . . . . . 13  |-  ( A  e.  ( -u pi (,] pi )  ->  -u pi  <  A )
8180adantr 274 . . . . . . . . . . . 12  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  -u pi  <  A )
8279, 8, 81ltnegcon1d 8294 . . . . . . . . . . 11  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  -u A  <  pi )
8382adantr 274 . . . . . . . . . 10  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  -u A  < 
pi )
8470, 78, 83ltled 7888 . . . . . . . . 9  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  -u A  <_  pi )
8571, 2elicc2i 9729 . . . . . . . . 9  |-  ( -u A  e.  ( 0 [,] pi )  <->  ( -u A  e.  RR  /\  0  <_  -u A  /\  -u A  <_  pi ) )
8670, 77, 84, 85syl3anbrc 1165 . . . . . . . 8  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  -u A  e.  ( 0 [,] pi ) )
87 coseq00topi 12926 . . . . . . . 8  |-  ( -u A  e.  ( 0 [,] pi )  -> 
( ( cos `  -u A
)  =  0  <->  -u A  =  ( pi  / 
2 ) ) )
8886, 87syl 14 . . . . . . 7  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  ( ( cos `  -u A )  =  0  <->  -u A  =  ( pi  /  2 ) ) )
8968, 88mpbid 146 . . . . . 6  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  -u A  =  ( pi  /  2
) )
9064, 89negcon1ad 8075 . . . . 5  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  -u ( pi 
/  2 )  =  A )
9190eqcomd 2145 . . . 4  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  A  =  -u ( pi  /  2
) )
92 prid2g 3628 . . . . 5  |-  ( -u ( pi  /  2
)  e.  RR  ->  -u ( pi  /  2
)  e.  { ( pi  /  2 ) ,  -u ( pi  / 
2 ) } )
93 eleq1a 2211 . . . . 5  |-  ( -u ( pi  /  2
)  e.  { ( pi  /  2 ) ,  -u ( pi  / 
2 ) }  ->  ( A  =  -u (
pi  /  2 )  ->  A  e.  {
( pi  /  2
) ,  -u (
pi  /  2 ) } ) )
9411, 92, 93mp2b 8 . . . 4  |-  ( A  =  -u ( pi  / 
2 )  ->  A  e.  { ( pi  / 
2 ) ,  -u ( pi  /  2
) } )
9591, 94syl 14 . . 3  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  0
)  ->  A  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )
96 pirp 12880 . . . . . . 7  |-  pi  e.  RR+
9713, 22elrpii 9451 . . . . . . 7  |-  4  e.  RR+
98 rpdivcl 9474 . . . . . . 7  |-  ( ( pi  e.  RR+  /\  4  e.  RR+ )  ->  (
pi  /  4 )  e.  RR+ )
9996, 97, 98mp2an 422 . . . . . 6  |-  ( pi 
/  4 )  e.  RR+
100 rpgt0 9460 . . . . . 6  |-  ( ( pi  /  4 )  e.  RR+  ->  0  < 
( pi  /  4
) )
10199, 100ax-mp 5 . . . . 5  |-  0  <  ( pi  /  4
)
102 lt0neg2 8238 . . . . . 6  |-  ( ( pi  /  4 )  e.  RR  ->  (
0  <  ( pi  /  4 )  <->  -u ( pi 
/  4 )  <  0 ) )
10315, 102ax-mp 5 . . . . 5  |-  ( 0  <  ( pi  / 
4 )  <->  -u ( pi 
/  4 )  <  0 )
104101, 103mpbi 144 . . . 4  |-  -u (
pi  /  4 )  <  0
105 axltwlin 7839 . . . . 5  |-  ( (
-u ( pi  / 
4 )  e.  RR  /\  0  e.  RR  /\  A  e.  RR )  ->  ( -u ( pi 
/  4 )  <  0  ->  ( -u (
pi  /  4 )  <  A  \/  A  <  0 ) ) )
10616, 71, 8, 105mp3an12i 1319 . . . 4  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  -> 
( -u ( pi  / 
4 )  <  0  ->  ( -u ( pi 
/  4 )  < 
A  \/  A  <  0 ) ) )
107104, 106mpi 15 . . 3  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  -> 
( -u ( pi  / 
4 )  <  A  \/  A  <  0
) )
10862, 95, 107mpjaodan 787 . 2  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  A  e.  { (
pi  /  2 ) ,  -u ( pi  / 
2 ) } )
109 elpri 3550 . . . 4  |-  ( A  e.  { ( pi 
/  2 ) , 
-u ( pi  / 
2 ) }  ->  ( A  =  ( pi 
/  2 )  \/  A  =  -u (
pi  /  2 ) ) )
110 fveq2 5421 . . . . . 6  |-  ( A  =  ( pi  / 
2 )  ->  ( cos `  A )  =  ( cos `  (
pi  /  2 ) ) )
111 coshalfpi 12888 . . . . . 6  |-  ( cos `  ( pi  /  2
) )  =  0
112110, 111syl6eq 2188 . . . . 5  |-  ( A  =  ( pi  / 
2 )  ->  ( cos `  A )  =  0 )
113 fveq2 5421 . . . . . 6  |-  ( A  =  -u ( pi  / 
2 )  ->  ( cos `  A )  =  ( cos `  -u (
pi  /  2 ) ) )
114 cosneghalfpi 12889 . . . . . 6  |-  ( cos `  -u ( pi  / 
2 ) )  =  0
115113, 114syl6eq 2188 . . . . 5  |-  ( A  =  -u ( pi  / 
2 )  ->  ( cos `  A )  =  0 )
116112, 115jaoi 705 . . . 4  |-  ( ( A  =  ( pi 
/  2 )  \/  A  =  -u (
pi  /  2 ) )  ->  ( cos `  A )  =  0 )
117109, 116syl 14 . . 3  |-  ( A  e.  { ( pi 
/  2 ) , 
-u ( pi  / 
2 ) }  ->  ( cos `  A )  =  0 )
118117adantl 275 . 2  |-  ( ( A  e.  ( -u pi (,] pi )  /\  A  e.  { (
pi  /  2 ) ,  -u ( pi  / 
2 ) } )  ->  ( cos `  A
)  =  0 )
119108, 118impbida 585 1  |-  ( A  e.  ( -u pi (,] pi )  ->  (
( cos `  A
)  =  0  <->  A  e.  { ( pi  / 
2 ) ,  -u ( pi  /  2
) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    /\ w3a 962    = wceq 1331    e. wcel 1480   {cpr 3528   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7625   RRcr 7626   0cc0 7627    x. cmul 7632   RR*cxr 7806    < clt 7807    <_ cle 7808   -ucneg 7941    / cdiv 8439   2c2 8778   3c3 8779   4c4 8780   RR+crp 9448   (,)cioo 9678   (,]cioc 9679   [,]cicc 9681   cosccos 11358   picpi 11360
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  ax-pre-suploc 7748  ax-addf 7749  ax-mulf 7750
This theorem depends on definitions:  df-bi 116  df-stab 816  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-of 5982  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-map 6544  df-pm 6545  df-en 6635  df-dom 6636  df-fin 6637  df-sup 6871  df-inf 6872  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-9 8793  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-xneg 9566  df-xadd 9567  df-ioo 9682  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  df-pi 11366  df-rest 12132  df-topgen 12151  df-psmet 12166  df-xmet 12167  df-met 12168  df-bl 12169  df-mopn 12170  df-top 12175  df-topon 12188  df-bases 12220  df-ntr 12275  df-cn 12367  df-cnp 12368  df-tx 12432  df-cncf 12737  df-limced 12804  df-dvap 12805
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator