MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2efiatan Structured version   Visualization version   Unicode version

Theorem 2efiatan 23844
Description: Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 2-Apr-2015.)
Assertion
Ref Expression
2efiatan  |-  ( A  e.  dom arctan  ->  ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 ) )

Proof of Theorem 2efiatan
StepHypRef Expression
1 atanval 23810 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
21oveq2d 6306 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  (arctan `  A
) )  =  ( ( 2  x.  _i )  x.  ( (
_i  /  2 )  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) ) ) )
3 2cn 10680 . . . . . 6  |-  2  e.  CC
43a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  2  e.  CC )
5 ax-icn 9598 . . . . . 6  |-  _i  e.  CC
65a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
7 atancl 23807 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
84, 6, 7mulassd 9666 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  (arctan `  A
) )  =  ( 2  x.  ( _i  x.  (arctan `  A
) ) ) )
9 halfcl 10838 . . . . . . . . . 10  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
105, 9ax-mp 5 . . . . . . . . 9  |-  ( _i 
/  2 )  e.  CC
113, 5, 10mulassi 9652 . . . . . . . 8  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  =  ( 2  x.  (
_i  x.  ( _i  /  2 ) ) )
123, 5, 10mul12i 9828 . . . . . . . 8  |-  ( 2  x.  ( _i  x.  ( _i  /  2
) ) )  =  ( _i  x.  (
2  x.  ( _i 
/  2 ) ) )
13 2ne0 10702 . . . . . . . . . . 11  |-  2  =/=  0
145, 3, 13divcan2i 10350 . . . . . . . . . 10  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
1514oveq2i 6301 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  =  ( _i  x.  _i )
16 ixi 10241 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
1715, 16eqtri 2473 . . . . . . . 8  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  = 
-u 1
1811, 12, 173eqtri 2477 . . . . . . 7  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  = 
-u 1
1918oveq1i 6300 . . . . . 6  |-  ( ( ( 2  x.  _i )  x.  ( _i  /  2 ) )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  =  ( -u 1  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )
20 ax-1cn 9597 . . . . . . . . . 10  |-  1  e.  CC
21 atandm2 23803 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
2221simp1bi 1023 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  A  e.  CC )
23 mulcl 9623 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
245, 22, 23sylancr 669 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( _i  x.  A )  e.  CC )
25 subcl 9874 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
2620, 24, 25sylancr 669 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
2721simp2bi 1024 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
2826, 27logcld 23520 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
29 addcl 9621 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
3020, 24, 29sylancr 669 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
3121simp3bi 1025 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
3230, 31logcld 23520 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
3328, 32subcld 9986 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
3433mulm1d 10070 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( -u
1  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) )  = 
-u ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )
3519, 34syl5eq 2497 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  x.  ( _i  /  2 ) )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  =  -u ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )
36 2mulicn 10836 . . . . . . 7  |-  ( 2  x.  _i )  e.  CC
3736a1i 11 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  e.  CC )
3810a1i 11 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i 
/  2 )  e.  CC )
3937, 38, 33mulassd 9666 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  x.  ( _i  /  2 ) )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  =  ( ( 2  x.  _i )  x.  ( ( _i  / 
2 )  x.  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) ) )
4028, 32negsubdi2d 10002 . . . . 5  |-  ( A  e.  dom arctan  ->  -u (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )
4135, 39, 403eqtr3d 2493 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  ( ( _i 
/  2 )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
422, 8, 413eqtr3d 2493 . . 3  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  x.  (arctan `  A ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
4342fveq2d 5869 . 2  |-  ( A  e.  dom arctan  ->  ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  =  ( exp `  (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
44 efsub 14154 . . 3  |-  ( ( ( log `  (
1  +  ( _i  x.  A ) ) )  e.  CC  /\  ( log `  ( 1  -  ( _i  x.  A ) ) )  e.  CC )  -> 
( exp `  (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) )  /  ( exp `  ( log `  (
1  -  ( _i  x.  A ) ) ) ) ) )
4532, 28, 44syl2anc 667 . 2  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) )  /  ( exp `  ( log `  (
1  -  ( _i  x.  A ) ) ) ) ) )
46 eflog 23526 . . . . 5  |-  ( ( ( 1  +  ( _i  x.  A ) )  e.  CC  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 )  ->  ( exp `  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  =  ( 1  +  ( _i  x.  A
) ) )
4730, 31, 46syl2anc 667 . . . 4  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) )  =  ( 1  +  ( _i  x.  A ) ) )
48 eflog 23526 . . . . 5  |-  ( ( ( 1  -  (
_i  x.  A )
)  e.  CC  /\  ( 1  -  (
_i  x.  A )
)  =/=  0 )  ->  ( exp `  ( log `  ( 1  -  ( _i  x.  A
) ) ) )  =  ( 1  -  ( _i  x.  A
) ) )
4926, 27, 48syl2anc 667 . . . 4  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  -  ( _i  x.  A ) ) ) )  =  ( 1  -  ( _i  x.  A ) ) )
5047, 49oveq12d 6308 . . 3  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  / 
( exp `  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( ( 1  +  ( _i  x.  A ) )  /  ( 1  -  ( _i  x.  A
) ) ) )
51 negsub 9922 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  +  -u A )  =  ( _i  -  A ) )
525, 22, 51sylancr 669 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( _i  +  -u A )  =  ( _i  -  A
) )
536mulid1d 9660 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( _i  x.  1 )  =  _i )
5416oveq1i 6300 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  A )  =  ( -u 1  x.  A )
556, 6, 22mulassd 9666 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  _i )  x.  A )  =  ( _i  x.  (
_i  x.  A )
) )
5622mulm1d 10070 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( -u
1  x.  A )  =  -u A )
5754, 55, 563eqtr3a 2509 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( _i  x.  A ) )  = 
-u A )
5853, 57oveq12d 6308 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  +  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  +  -u A ) )
596, 22, 6pnpcan2d 10024 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  +  _i )  -  ( A  +  _i ) )  =  ( _i  -  A ) )
6052, 58, 593eqtr4d 2495 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  +  ( _i  x.  ( _i  x.  A
) ) )  =  ( ( _i  +  _i )  -  ( A  +  _i )
) )
6120a1i 11 . . . . . . 7  |-  ( A  e.  dom arctan  ->  1  e.  CC )
626, 61, 24adddid 9667 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  +  ( _i  x.  A
) ) )  =  ( ( _i  x.  1 )  +  ( _i  x.  ( _i  x.  A ) ) ) )
6362timesd 10855 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  =  ( _i  +  _i ) )
6463oveq1d 6305 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  -  ( A  +  _i ) )  =  ( ( _i  +  _i )  -  ( A  +  _i ) ) )
6560, 62, 643eqtr4d 2495 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  +  ( _i  x.  A
) ) )  =  ( ( 2  x.  _i )  -  ( A  +  _i )
) )
666, 61, 24subdid 10074 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( ( _i  x.  1 )  -  (
_i  x.  ( _i  x.  A ) ) ) )
6753, 57oveq12d 6308 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  -  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  -  -u A
) )
68 subneg 9923 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  -  -u A
)  =  ( _i  +  A ) )
695, 22, 68sylancr 669 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( _i 
-  -u A )  =  ( _i  +  A
) )
7067, 69eqtrd 2485 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  1 )  -  ( _i  x.  ( _i  x.  A
) ) )  =  ( _i  +  A
) )
71 addcom 9819 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  +  A
)  =  ( A  +  _i ) )
725, 22, 71sylancr 669 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( _i  +  A )  =  ( A  +  _i ) )
7366, 70, 723eqtrd 2489 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( A  +  _i ) )
7465, 73oveq12d 6308 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  ( 1  +  ( _i  x.  A ) ) )  /  ( _i  x.  ( 1  -  (
_i  x.  A )
) ) )  =  ( ( ( 2  x.  _i )  -  ( A  +  _i ) )  /  ( A  +  _i )
) )
75 ine0 10054 . . . . . 6  |-  _i  =/=  0
7675a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  _i  =/=  0 )
7730, 26, 6, 27, 76divcan5d 10409 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  ( 1  +  ( _i  x.  A ) ) )  /  ( _i  x.  ( 1  -  (
_i  x.  A )
) ) )  =  ( ( 1  +  ( _i  x.  A
) )  /  (
1  -  ( _i  x.  A ) ) ) )
78 addcl 9621 . . . . . 6  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  +  _i )  e.  CC )
7922, 5, 78sylancl 668 . . . . 5  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  e.  CC )
80 subneg 9923 . . . . . . 7  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  -  -u _i )  =  ( A  +  _i ) )
8122, 5, 80sylancl 668 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =  ( A  +  _i ) )
82 atandm 23802 . . . . . . . 8  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
8382simp2bi 1024 . . . . . . 7  |-  ( A  e.  dom arctan  ->  A  =/=  -u _i )
84 negicn 9876 . . . . . . . 8  |-  -u _i  e.  CC
85 subeq0 9900 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =  0  <->  A  =  -u _i ) )
8685necon3bid 2668 . . . . . . . 8  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
8722, 84, 86sylancl 668 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
8883, 87mpbird 236 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =/=  0 )
8981, 88eqnetrrd 2692 . . . . 5  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  =/=  0 )
9037, 79, 79, 89divsubdird 10422 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( A  +  _i ) )  / 
( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  ( ( A  +  _i )  /  ( A  +  _i ) ) ) )
9174, 77, 903eqtr3d 2493 . . 3  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  /  ( 1  -  ( _i  x.  A
) ) )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  (
( A  +  _i )  /  ( A  +  _i ) ) ) )
9279, 89dividd 10381 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( A  +  _i )  /  ( A  +  _i ) )  =  1 )
9392oveq2d 6306 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
( A  +  _i )  /  ( A  +  _i ) ) )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  1 ) )
9450, 91, 933eqtrd 2489 . 2  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  / 
( exp `  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 ) )
9543, 45, 943eqtrd 2489 1  |-  ( A  e.  dom arctan  ->  ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   dom cdm 4834   ` cfv 5582  (class class class)co 6290   CCcc 9537   0cc0 9539   1c1 9540   _ici 9541    + caddc 9542    x. cmul 9544    - cmin 9860   -ucneg 9861    / cdiv 10269   2c2 10659   expce 14114   logclog 23504  arctancatan 23790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124  df-pi 14126  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-log 23506  df-atan 23793
This theorem is referenced by:  tanatan  23845
  Copyright terms: Public domain W3C validator