Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvacos Structured version   Visualization version   Unicode version

Theorem dvacos 32093
Description: Derivative of arccosine. (Contributed by Brendan Leahy, 18-Dec-2018.)
Hypothesis
Ref Expression
dvasin.d  |-  D  =  ( CC  \  (
( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) ) )
Assertion
Ref Expression
dvacos  |-  ( CC 
_D  (arccos  |`  D ) )  =  ( x  e.  D  |->  ( -u
1  /  ( sqr `  ( 1  -  (
x ^ 2 ) ) ) ) )
Distinct variable group:    x, D

Proof of Theorem dvacos
StepHypRef Expression
1 df-acos 23871 . . . . 5  |- arccos  =  ( x  e.  CC  |->  ( ( pi  /  2
)  -  (arcsin `  x ) ) )
21reseq1i 5107 . . . 4  |-  (arccos  |`  D )  =  ( ( x  e.  CC  |->  ( ( pi  /  2 )  -  (arcsin `  x
) ) )  |`  D )
3 dvasin.d . . . . . 6  |-  D  =  ( CC  \  (
( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) ) )
4 difss 3549 . . . . . 6  |-  ( CC 
\  ( ( -oo (,] -u 1 )  u.  ( 1 [,) +oo ) ) )  C_  CC
53, 4eqsstri 3448 . . . . 5  |-  D  C_  CC
6 resmpt 5160 . . . . 5  |-  ( D 
C_  CC  ->  ( ( x  e.  CC  |->  ( ( pi  /  2
)  -  (arcsin `  x ) ) )  |`  D )  =  ( x  e.  D  |->  ( ( pi  /  2
)  -  (arcsin `  x ) ) ) )
75, 6ax-mp 5 . . . 4  |-  ( ( x  e.  CC  |->  ( ( pi  /  2
)  -  (arcsin `  x ) ) )  |`  D )  =  ( x  e.  D  |->  ( ( pi  /  2
)  -  (arcsin `  x ) ) )
82, 7eqtri 2493 . . 3  |-  (arccos  |`  D )  =  ( x  e.  D  |->  ( ( pi 
/  2 )  -  (arcsin `  x ) ) )
98oveq2i 6319 . 2  |-  ( CC 
_D  (arccos  |`  D ) )  =  ( CC 
_D  ( x  e.  D  |->  ( ( pi 
/  2 )  -  (arcsin `  x ) ) ) )
10 cnelprrecn 9650 . . . . 5  |-  CC  e.  { RR ,  CC }
1110a1i 11 . . . 4  |-  ( T. 
->  CC  e.  { RR ,  CC } )
12 halfpire 23498 . . . . . 6  |-  ( pi 
/  2 )  e.  RR
1312recni 9673 . . . . 5  |-  ( pi 
/  2 )  e.  CC
1413a1i 11 . . . 4  |-  ( ( T.  /\  x  e.  D )  ->  (
pi  /  2 )  e.  CC )
15 c0ex 9655 . . . . 5  |-  0  e.  _V
1615a1i 11 . . . 4  |-  ( ( T.  /\  x  e.  D )  ->  0  e.  _V )
1713a1i 11 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  (
pi  /  2 )  e.  CC )
1815a1i 11 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  0  e.  _V )
1913a1i 11 . . . . . 6  |-  ( T. 
->  ( pi  /  2
)  e.  CC )
2011, 19dvmptc 22991 . . . . 5  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( pi  /  2 ) ) )  =  ( x  e.  CC  |->  0 ) )
215a1i 11 . . . . 5  |-  ( T. 
->  D  C_  CC )
22 eqid 2471 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2322cnfldtopon 21881 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
2423toponunii 20024 . . . . . . . 8  |-  CC  =  U. ( TopOpen ` fld )
2524restid 15410 . . . . . . 7  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
2623, 25ax-mp 5 . . . . . 6  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
2726eqcomi 2480 . . . . 5  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
2822recld2 21910 . . . . . . . . . 10  |-  RR  e.  ( Clsd `  ( TopOpen ` fld ) )
29 neg1rr 10736 . . . . . . . . . . . 12  |-  -u 1  e.  RR
30 iocmnfcld 21867 . . . . . . . . . . . 12  |-  ( -u
1  e.  RR  ->  ( -oo (,] -u 1
)  e.  ( Clsd `  ( topGen `  ran  (,) )
) )
3129, 30ax-mp 5 . . . . . . . . . . 11  |-  ( -oo (,] -u 1 )  e.  ( Clsd `  ( topGen `
 ran  (,) )
)
3222tgioo2 21899 . . . . . . . . . . . 12  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
3332fveq2i 5882 . . . . . . . . . . 11  |-  ( Clsd `  ( topGen `  ran  (,) )
)  =  ( Clsd `  ( ( TopOpen ` fld )t  RR ) )
3431, 33eleqtri 2547 . . . . . . . . . 10  |-  ( -oo (,] -u 1 )  e.  ( Clsd `  (
( TopOpen ` fld )t  RR ) )
35 restcldr 20267 . . . . . . . . . 10  |-  ( ( RR  e.  ( Clsd `  ( TopOpen ` fld ) )  /\  ( -oo (,] -u 1 )  e.  ( Clsd `  (
( TopOpen ` fld )t  RR ) ) )  ->  ( -oo (,] -u 1 )  e.  (
Clsd `  ( TopOpen ` fld ) ) )
3628, 34, 35mp2an 686 . . . . . . . . 9  |-  ( -oo (,] -u 1 )  e.  ( Clsd `  ( TopOpen
` fld
) )
37 1re 9660 . . . . . . . . . . . 12  |-  1  e.  RR
38 icopnfcld 21866 . . . . . . . . . . . 12  |-  ( 1  e.  RR  ->  (
1 [,) +oo )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
3937, 38ax-mp 5 . . . . . . . . . . 11  |-  ( 1 [,) +oo )  e.  ( Clsd `  ( topGen `
 ran  (,) )
)
4039, 33eleqtri 2547 . . . . . . . . . 10  |-  ( 1 [,) +oo )  e.  ( Clsd `  (
( TopOpen ` fld )t  RR ) )
41 restcldr 20267 . . . . . . . . . 10  |-  ( ( RR  e.  ( Clsd `  ( TopOpen ` fld ) )  /\  (
1 [,) +oo )  e.  ( Clsd `  (
( TopOpen ` fld )t  RR ) ) )  ->  ( 1 [,) +oo )  e.  ( Clsd `  ( TopOpen ` fld ) ) )
4228, 40, 41mp2an 686 . . . . . . . . 9  |-  ( 1 [,) +oo )  e.  ( Clsd `  ( TopOpen
` fld
) )
43 uncld 20133 . . . . . . . . 9  |-  ( ( ( -oo (,] -u 1
)  e.  ( Clsd `  ( TopOpen ` fld ) )  /\  (
1 [,) +oo )  e.  ( Clsd `  ( TopOpen
` fld
) ) )  -> 
( ( -oo (,] -u 1 )  u.  (
1 [,) +oo )
)  e.  ( Clsd `  ( TopOpen ` fld ) ) )
4436, 42, 43mp2an 686 . . . . . . . 8  |-  ( ( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) )  e.  ( Clsd `  ( TopOpen
` fld
) )
4524cldopn 20123 . . . . . . . 8  |-  ( ( ( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) )  e.  ( Clsd `  ( TopOpen
` fld
) )  ->  ( CC  \  ( ( -oo (,] -u 1 )  u.  ( 1 [,) +oo ) ) )  e.  ( TopOpen ` fld ) )
4644, 45ax-mp 5 . . . . . . 7  |-  ( CC 
\  ( ( -oo (,] -u 1 )  u.  ( 1 [,) +oo ) ) )  e.  ( TopOpen ` fld )
473, 46eqeltri 2545 . . . . . 6  |-  D  e.  ( TopOpen ` fld )
4847a1i 11 . . . . 5  |-  ( T. 
->  D  e.  ( TopOpen
` fld
) )
4911, 17, 18, 20, 21, 27, 22, 48dvmptres 22996 . . . 4  |-  ( T. 
->  ( CC  _D  (
x  e.  D  |->  ( pi  /  2 ) ) )  =  ( x  e.  D  |->  0 ) )
505sseli 3414 . . . . . 6  |-  ( x  e.  D  ->  x  e.  CC )
51 asincl 23878 . . . . . 6  |-  ( x  e.  CC  ->  (arcsin `  x )  e.  CC )
5250, 51syl 17 . . . . 5  |-  ( x  e.  D  ->  (arcsin `  x )  e.  CC )
5352adantl 473 . . . 4  |-  ( ( T.  /\  x  e.  D )  ->  (arcsin `  x )  e.  CC )
54 ovex 6336 . . . . 5  |-  ( 1  /  ( sqr `  (
1  -  ( x ^ 2 ) ) ) )  e.  _V
5554a1i 11 . . . 4  |-  ( ( T.  /\  x  e.  D )  ->  (
1  /  ( sqr `  ( 1  -  (
x ^ 2 ) ) ) )  e. 
_V )
563dvasin 32092 . . . . 5  |-  ( CC 
_D  (arcsin  |`  D ) )  =  ( x  e.  D  |->  ( 1  /  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) )
57 asinf 23877 . . . . . . . 8  |- arcsin : CC --> CC
5857a1i 11 . . . . . . 7  |-  ( T. 
-> arcsin : CC --> CC )
5958, 21feqresmpt 5933 . . . . . 6  |-  ( T. 
->  (arcsin  |`  D )  =  ( x  e.  D  |->  (arcsin `  x )
) )
6059oveq2d 6324 . . . . 5  |-  ( T. 
->  ( CC  _D  (arcsin  |`  D ) )  =  ( CC  _D  (
x  e.  D  |->  (arcsin `  x ) ) ) )
6156, 60syl5reqr 2520 . . . 4  |-  ( T. 
->  ( CC  _D  (
x  e.  D  |->  (arcsin `  x ) ) )  =  ( x  e.  D  |->  ( 1  / 
( sqr `  (
1  -  ( x ^ 2 ) ) ) ) ) )
6211, 14, 16, 49, 53, 55, 61dvmptsub 23000 . . 3  |-  ( T. 
->  ( CC  _D  (
x  e.  D  |->  ( ( pi  /  2
)  -  (arcsin `  x ) ) ) )  =  ( x  e.  D  |->  ( 0  -  ( 1  / 
( sqr `  (
1  -  ( x ^ 2 ) ) ) ) ) ) )
6362trud 1461 . 2  |-  ( CC 
_D  ( x  e.  D  |->  ( ( pi 
/  2 )  -  (arcsin `  x ) ) ) )  =  ( x  e.  D  |->  ( 0  -  ( 1  /  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) ) )
64 df-neg 9883 . . . 4  |-  -u (
1  /  ( sqr `  ( 1  -  (
x ^ 2 ) ) ) )  =  ( 0  -  (
1  /  ( sqr `  ( 1  -  (
x ^ 2 ) ) ) ) )
65 1cnd 9677 . . . . 5  |-  ( x  e.  D  ->  1  e.  CC )
66 ax-1cn 9615 . . . . . . 7  |-  1  e.  CC
6750sqcld 12452 . . . . . . 7  |-  ( x  e.  D  ->  (
x ^ 2 )  e.  CC )
68 subcl 9894 . . . . . . 7  |-  ( ( 1  e.  CC  /\  ( x ^ 2 )  e.  CC )  ->  ( 1  -  ( x ^ 2 ) )  e.  CC )
6966, 67, 68sylancr 676 . . . . . 6  |-  ( x  e.  D  ->  (
1  -  ( x ^ 2 ) )  e.  CC )
7069sqrtcld 13576 . . . . 5  |-  ( x  e.  D  ->  ( sqr `  ( 1  -  ( x ^ 2 ) ) )  e.  CC )
71 eldifn 3545 . . . . . . . 8  |-  ( x  e.  ( CC  \ 
( ( -oo (,] -u 1 )  u.  (
1 [,) +oo )
) )  ->  -.  x  e.  ( ( -oo (,] -u 1 )  u.  ( 1 [,) +oo ) ) )
7271, 3eleq2s 2567 . . . . . . 7  |-  ( x  e.  D  ->  -.  x  e.  ( ( -oo (,] -u 1 )  u.  ( 1 [,) +oo ) ) )
73 mnfxr 11437 . . . . . . . . . . . 12  |- -oo  e.  RR*
7429rexri 9711 . . . . . . . . . . . 12  |-  -u 1  e.  RR*
75 mnflt 11448 . . . . . . . . . . . . 13  |-  ( -u
1  e.  RR  -> -oo 
<  -u 1 )
7629, 75ax-mp 5 . . . . . . . . . . . 12  |- -oo  <  -u 1
77 ubioc1 11713 . . . . . . . . . . . 12  |-  ( ( -oo  e.  RR*  /\  -u 1  e.  RR*  /\ -oo  <  -u 1 )  ->  -u 1  e.  ( -oo (,] -u 1
) )
7873, 74, 76, 77mp3an 1390 . . . . . . . . . . 11  |-  -u 1  e.  ( -oo (,] -u 1
)
79 eleq1 2537 . . . . . . . . . . 11  |-  ( x  =  -u 1  ->  (
x  e.  ( -oo (,] -u 1 )  <->  -u 1  e.  ( -oo (,] -u 1
) ) )
8078, 79mpbiri 241 . . . . . . . . . 10  |-  ( x  =  -u 1  ->  x  e.  ( -oo (,] -u 1
) )
8137rexri 9711 . . . . . . . . . . . 12  |-  1  e.  RR*
82 pnfxr 11435 . . . . . . . . . . . 12  |- +oo  e.  RR*
83 ltpnf 11445 . . . . . . . . . . . . 13  |-  ( 1  e.  RR  ->  1  < +oo )
8437, 83ax-mp 5 . . . . . . . . . . . 12  |-  1  < +oo
85 lbico1 11714 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR*  /\ +oo  e.  RR*  /\  1  < +oo )  ->  1  e.  ( 1 [,) +oo ) )
8681, 82, 84, 85mp3an 1390 . . . . . . . . . . 11  |-  1  e.  ( 1 [,) +oo )
87 eleq1 2537 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  e.  ( 1 [,) +oo )  <->  1  e.  ( 1 [,) +oo ) ) )
8886, 87mpbiri 241 . . . . . . . . . 10  |-  ( x  =  1  ->  x  e.  ( 1 [,) +oo ) )
8980, 88orim12i 525 . . . . . . . . 9  |-  ( ( x  =  -u 1  \/  x  =  1
)  ->  ( x  e.  ( -oo (,] -u 1
)  \/  x  e.  ( 1 [,) +oo ) ) )
9089orcoms 396 . . . . . . . 8  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  ( x  e.  ( -oo (,] -u 1
)  \/  x  e.  ( 1 [,) +oo ) ) )
91 elun 3565 . . . . . . . 8  |-  ( x  e.  ( ( -oo (,] -u 1 )  u.  ( 1 [,) +oo ) )  <->  ( x  e.  ( -oo (,] -u 1
)  \/  x  e.  ( 1 [,) +oo ) ) )
9290, 91sylibr 217 . . . . . . 7  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  x  e.  ( ( -oo (,] -u 1 )  u.  (
1 [,) +oo )
) )
9372, 92nsyl 125 . . . . . 6  |-  ( x  e.  D  ->  -.  ( x  =  1  \/  x  =  -u 1
) )
94 sq1 12407 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
95 1cnd 9677 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  ( sqr `  ( 1  -  ( x ^
2 ) ) )  =  0 )  -> 
1  e.  CC )
96 sqcl 12375 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x ^ 2 )  e.  CC )
9796adantr 472 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  ( sqr `  ( 1  -  ( x ^
2 ) ) )  =  0 )  -> 
( x ^ 2 )  e.  CC )
9866, 96, 68sylancr 676 . . . . . . . . . . . . 13  |-  ( x  e.  CC  ->  (
1  -  ( x ^ 2 ) )  e.  CC )
9998adantr 472 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  ( sqr `  ( 1  -  ( x ^
2 ) ) )  =  0 )  -> 
( 1  -  (
x ^ 2 ) )  e.  CC )
100 simpr 468 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  ( sqr `  ( 1  -  ( x ^
2 ) ) )  =  0 )  -> 
( sqr `  (
1  -  ( x ^ 2 ) ) )  =  0 )
10199, 100sqr00d 13580 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  ( sqr `  ( 1  -  ( x ^
2 ) ) )  =  0 )  -> 
( 1  -  (
x ^ 2 ) )  =  0 )
10295, 97, 101subeq0d 10013 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  ( sqr `  ( 1  -  ( x ^
2 ) ) )  =  0 )  -> 
1  =  ( x ^ 2 ) )
10394, 102syl5req 2518 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  ( sqr `  ( 1  -  ( x ^
2 ) ) )  =  0 )  -> 
( x ^ 2 )  =  ( 1 ^ 2 ) )
104103ex 441 . . . . . . . 8  |-  ( x  e.  CC  ->  (
( sqr `  (
1  -  ( x ^ 2 ) ) )  =  0  -> 
( x ^ 2 )  =  ( 1 ^ 2 ) ) )
105 sqeqor 12426 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  1  e.  CC )  ->  ( ( x ^
2 )  =  ( 1 ^ 2 )  <-> 
( x  =  1  \/  x  =  -u
1 ) ) )
10666, 105mpan2 685 . . . . . . . 8  |-  ( x  e.  CC  ->  (
( x ^ 2 )  =  ( 1 ^ 2 )  <->  ( x  =  1  \/  x  =  -u 1 ) ) )
107104, 106sylibd 222 . . . . . . 7  |-  ( x  e.  CC  ->  (
( sqr `  (
1  -  ( x ^ 2 ) ) )  =  0  -> 
( x  =  1  \/  x  =  -u
1 ) ) )
108107necon3bd 2657 . . . . . 6  |-  ( x  e.  CC  ->  ( -.  ( x  =  1  \/  x  =  -u
1 )  ->  ( sqr `  ( 1  -  ( x ^ 2 ) ) )  =/=  0 ) )
10950, 93, 108sylc 61 . . . . 5  |-  ( x  e.  D  ->  ( sqr `  ( 1  -  ( x ^ 2 ) ) )  =/=  0 )
11065, 70, 109divnegd 10418 . . . 4  |-  ( x  e.  D  ->  -u (
1  /  ( sqr `  ( 1  -  (
x ^ 2 ) ) ) )  =  ( -u 1  / 
( sqr `  (
1  -  ( x ^ 2 ) ) ) ) )
11164, 110syl5eqr 2519 . . 3  |-  ( x  e.  D  ->  (
0  -  ( 1  /  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) )  =  ( -u 1  / 
( sqr `  (
1  -  ( x ^ 2 ) ) ) ) )
112111mpteq2ia 4478 . 2  |-  ( x  e.  D  |->  ( 0  -  ( 1  / 
( sqr `  (
1  -  ( x ^ 2 ) ) ) ) ) )  =  ( x  e.  D  |->  ( -u 1  /  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) )
1139, 63, 1123eqtri 2497 1  |-  ( CC 
_D  (arccos  |`  D ) )  =  ( x  e.  D  |->  ( -u
1  /  ( sqr `  ( 1  -  (
x ^ 2 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452   T. wtru 1453    e. wcel 1904    =/= wne 2641   _Vcvv 3031    \ cdif 3387    u. cun 3388    C_ wss 3390   {cpr 3961   class class class wbr 4395    |-> cmpt 4454   ran crn 4840    |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692    < clt 9693    - cmin 9880   -ucneg 9881    / cdiv 10291   2c2 10681   (,)cioo 11660   (,]cioc 11661   [,)cico 11662   ^cexp 12310   sqrcsqrt 13373   picpi 14196   ↾t crest 15397   TopOpenctopn 15398   topGenctg 15414  ℂfldccnfld 19047  TopOnctopon 19995   Clsdccld 20108    _D cdv 22897  arcsincasin 23867  arccoscacos 23868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-tan 14202  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586  df-asin 23870  df-acos 23871
This theorem is referenced by:  dvreacos  32095
  Copyright terms: Public domain W3C validator