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Theorem dvacos 29670
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 22920 . . . . 5  |- arccos  =  ( x  e.  CC  |->  ( ( pi  /  2
)  -  (arcsin `  x ) ) )
21reseq1i 5262 . . . 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 3626 . . . . . 6  |-  ( CC 
\  ( ( -oo (,] -u 1 )  u.  ( 1 [,) +oo ) ) )  C_  CC
53, 4eqsstri 3529 . . . . 5  |-  D  C_  CC
6 resmpt 5316 . . . . 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 2491 . . 3  |-  (arccos  |`  D )  =  ( x  e.  D  |->  ( ( pi 
/  2 )  -  (arcsin `  x ) ) )
98oveq2i 6288 . 2  |-  ( CC 
_D  (arccos  |`  D ) )  =  ( CC 
_D  ( x  e.  D  |->  ( ( pi 
/  2 )  -  (arcsin `  x ) ) ) )
10 cnelprrecn 9576 . . . . 5  |-  CC  e.  { RR ,  CC }
1110a1i 11 . . . 4  |-  ( T. 
->  CC  e.  { RR ,  CC } )
12 halfpire 22585 . . . . . 6  |-  ( pi 
/  2 )  e.  RR
1312recni 9599 . . . . 5  |-  ( pi 
/  2 )  e.  CC
1413a1i 11 . . . 4  |-  ( ( T.  /\  x  e.  D )  ->  (
pi  /  2 )  e.  CC )
15 c0ex 9581 . . . . 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 22091 . . . . 5  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( pi  /  2 ) ) )  =  ( x  e.  CC  |->  0 ) )
215a1i 11 . . . . 5  |-  ( T. 
->  D  C_  CC )
22 eqid 2462 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2322cnfldtopon 21020 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
2423toponunii 19195 . . . . . . . 8  |-  CC  =  U. ( TopOpen ` fld )
2524restid 14680 . . . . . . 7  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
2623, 25ax-mp 5 . . . . . 6  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
2726eqcomi 2475 . . . . 5  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
2822recld2 21049 . . . . . . . . . 10  |-  RR  e.  ( Clsd `  ( TopOpen ` fld ) )
29 neg1rr 10631 . . . . . . . . . . . 12  |-  -u 1  e.  RR
30 iocmnfcld 21006 . . . . . . . . . . . 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 21038 . . . . . . . . . . . 12  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
3332fveq2i 5862 . . . . . . . . . . 11  |-  ( Clsd `  ( topGen `  ran  (,) )
)  =  ( Clsd `  ( ( TopOpen ` fld )t  RR ) )
3431, 33eleqtri 2548 . . . . . . . . . 10  |-  ( -oo (,] -u 1 )  e.  ( Clsd `  (
( TopOpen ` fld )t  RR ) )
35 restcldr 19436 . . . . . . . . . 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 672 . . . . . . . . 9  |-  ( -oo (,] -u 1 )  e.  ( Clsd `  ( TopOpen
` fld
) )
37 1re 9586 . . . . . . . . . . . 12  |-  1  e.  RR
38 icopnfcld 21005 . . . . . . . . . . . 12  |-  ( 1  e.  RR  ->  (
1 [,) +oo )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
3937, 38ax-mp 5 . . . . . . . . . . 11  |-  ( 1 [,) +oo )  e.  ( Clsd `  ( topGen `
 ran  (,) )
)
4039, 33eleqtri 2548 . . . . . . . . . 10  |-  ( 1 [,) +oo )  e.  ( Clsd `  (
( TopOpen ` fld )t  RR ) )
41 restcldr 19436 . . . . . . . . . 10  |-  ( ( RR  e.  ( Clsd `  ( TopOpen ` fld ) )  /\  (
1 [,) +oo )  e.  ( Clsd `  (
( TopOpen ` fld )t  RR ) ) )  ->  ( 1 [,) +oo )  e.  ( Clsd `  ( TopOpen ` fld ) ) )
4228, 40, 41mp2an 672 . . . . . . . . 9  |-  ( 1 [,) +oo )  e.  ( Clsd `  ( TopOpen
` fld
) )
43 uncld 19303 . . . . . . . . 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 672 . . . . . . . 8  |-  ( ( -oo (,] -u 1
)  u.  ( 1 [,) +oo ) )  e.  ( Clsd `  ( TopOpen
` fld
) )
4524cldopn 19293 . . . . . . . 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 2546 . . . . . 6  |-  D  e.  ( TopOpen ` fld )
4847a1i 11 . . . . 5  |-  ( T. 
->  D  e.  ( TopOpen
` fld
) )
4911, 17, 18, 20, 21, 27, 22, 48dvmptres 22096 . . . 4  |-  ( T. 
->  ( CC  _D  (
x  e.  D  |->  ( pi  /  2 ) ) )  =  ( x  e.  D  |->  0 ) )
505sseli 3495 . . . . . 6  |-  ( x  e.  D  ->  x  e.  CC )
51 asincl 22927 . . . . . 6  |-  ( x  e.  CC  ->  (arcsin `  x )  e.  CC )
5250, 51syl 16 . . . . 5  |-  ( x  e.  D  ->  (arcsin `  x )  e.  CC )
5352adantl 466 . . . 4  |-  ( ( T.  /\  x  e.  D )  ->  (arcsin `  x )  e.  CC )
54 ovex 6302 . . . . 5  |-  ( 1  /  ( sqr `  (
1  -  ( x ^ 2 ) ) ) )  e.  _V
5554a1i 11 . . . 4  |-  ( ( T.  /\  x  e.  D )  ->  (
1  /  ( sqr `  ( 1  -  (
x ^ 2 ) ) ) )  e. 
_V )
563dvasin 29669 . . . . 5  |-  ( CC 
_D  (arcsin  |`  D ) )  =  ( x  e.  D  |->  ( 1  /  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) )
57 asinf 22926 . . . . . . . 8  |- arcsin : CC --> CC
5857a1i 11 . . . . . . 7  |-  ( T. 
-> arcsin : CC --> CC )
5958, 21feqresmpt 5914 . . . . . 6  |-  ( T. 
->  (arcsin  |`  D )  =  ( x  e.  D  |->  (arcsin `  x )
) )
6059oveq2d 6293 . . . . 5  |-  ( T. 
->  ( CC  _D  (arcsin  |`  D ) )  =  ( CC  _D  (
x  e.  D  |->  (arcsin `  x ) ) ) )
6156, 60syl5reqr 2518 . . . 4  |-  ( T. 
->  ( CC  _D  (
x  e.  D  |->  (arcsin `  x ) ) )  =  ( x  e.  D  |->  ( 1  / 
( sqr `  (
1  -  ( x ^ 2 ) ) ) ) ) )
6211, 14, 16, 49, 53, 55, 61dvmptsub 22100 . . 3  |-  ( T. 
->  ( CC  _D  (
x  e.  D  |->  ( ( pi  /  2
)  -  (arcsin `  x ) ) ) )  =  ( x  e.  D  |->  ( 0  -  ( 1  / 
( sqr `  (
1  -  ( x ^ 2 ) ) ) ) ) ) )
6362trud 1383 . 2  |-  ( CC 
_D  ( x  e.  D  |->  ( ( pi 
/  2 )  -  (arcsin `  x ) ) ) )  =  ( x  e.  D  |->  ( 0  -  ( 1  /  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) ) )
64 df-neg 9799 . . . 4  |-  -u (
1  /  ( sqr `  ( 1  -  (
x ^ 2 ) ) ) )  =  ( 0  -  (
1  /  ( sqr `  ( 1  -  (
x ^ 2 ) ) ) ) )
65 1cnd 9603 . . . . 5  |-  ( x  e.  D  ->  1  e.  CC )
66 ax-1cn 9541 . . . . . . 7  |-  1  e.  CC
6750sqcld 12265 . . . . . . 7  |-  ( x  e.  D  ->  (
x ^ 2 )  e.  CC )
68 subcl 9810 . . . . . . 7  |-  ( ( 1  e.  CC  /\  ( x ^ 2 )  e.  CC )  ->  ( 1  -  ( x ^ 2 ) )  e.  CC )
6966, 67, 68sylancr 663 . . . . . 6  |-  ( x  e.  D  ->  (
1  -  ( x ^ 2 ) )  e.  CC )
7069sqrcld 13219 . . . . 5  |-  ( x  e.  D  ->  ( sqr `  ( 1  -  ( x ^ 2 ) ) )  e.  CC )
71 eldifn 3622 . . . . . . . 8  |-  ( x  e.  ( CC  \ 
( ( -oo (,] -u 1 )  u.  (
1 [,) +oo )
) )  ->  -.  x  e.  ( ( -oo (,] -u 1 )  u.  ( 1 [,) +oo ) ) )
7271, 3eleq2s 2570 . . . . . . 7  |-  ( x  e.  D  ->  -.  x  e.  ( ( -oo (,] -u 1 )  u.  ( 1 [,) +oo ) ) )
73 mnfxr 11314 . . . . . . . . . . . 12  |- -oo  e.  RR*
7429rexri 9637 . . . . . . . . . . . 12  |-  -u 1  e.  RR*
75 mnflt 11324 . . . . . . . . . . . . 13  |-  ( -u
1  e.  RR  -> -oo 
<  -u 1 )
7629, 75ax-mp 5 . . . . . . . . . . . 12  |- -oo  <  -u 1
77 ubioc1 11569 . . . . . . . . . . . 12  |-  ( ( -oo  e.  RR*  /\  -u 1  e.  RR*  /\ -oo  <  -u 1 )  ->  -u 1  e.  ( -oo (,] -u 1
) )
7873, 74, 76, 77mp3an 1319 . . . . . . . . . . 11  |-  -u 1  e.  ( -oo (,] -u 1
)
79 eleq1 2534 . . . . . . . . . . 11  |-  ( x  =  -u 1  ->  (
x  e.  ( -oo (,] -u 1 )  <->  -u 1  e.  ( -oo (,] -u 1
) ) )
8078, 79mpbiri 233 . . . . . . . . . 10  |-  ( x  =  -u 1  ->  x  e.  ( -oo (,] -u 1
) )
8137rexri 9637 . . . . . . . . . . . 12  |-  1  e.  RR*
82 pnfxr 11312 . . . . . . . . . . . 12  |- +oo  e.  RR*
83 ltpnf 11322 . . . . . . . . . . . . 13  |-  ( 1  e.  RR  ->  1  < +oo )
8437, 83ax-mp 5 . . . . . . . . . . . 12  |-  1  < +oo
85 lbico1 11570 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR*  /\ +oo  e.  RR*  /\  1  < +oo )  ->  1  e.  ( 1 [,) +oo ) )
8681, 82, 84, 85mp3an 1319 . . . . . . . . . . 11  |-  1  e.  ( 1 [,) +oo )
87 eleq1 2534 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  e.  ( 1 [,) +oo )  <->  1  e.  ( 1 [,) +oo ) ) )
8886, 87mpbiri 233 . . . . . . . . . 10  |-  ( x  =  1  ->  x  e.  ( 1 [,) +oo ) )
8980, 88orim12i 516 . . . . . . . . 9  |-  ( ( x  =  -u 1  \/  x  =  1
)  ->  ( x  e.  ( -oo (,] -u 1
)  \/  x  e.  ( 1 [,) +oo ) ) )
9089orcoms 389 . . . . . . . 8  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  ( x  e.  ( -oo (,] -u 1
)  \/  x  e.  ( 1 [,) +oo ) ) )
91 elun 3640 . . . . . . . 8  |-  ( x  e.  ( ( -oo (,] -u 1 )  u.  ( 1 [,) +oo ) )  <->  ( x  e.  ( -oo (,] -u 1
)  \/  x  e.  ( 1 [,) +oo ) ) )
9290, 91sylibr 212 . . . . . . 7  |-  ( ( x  =  1  \/  x  =  -u 1
)  ->  x  e.  ( ( -oo (,] -u 1 )  u.  (
1 [,) +oo )
) )
9372, 92nsyl 121 . . . . . 6  |-  ( x  e.  D  ->  -.  ( x  =  1  \/  x  =  -u 1
) )
94 sq1 12219 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
95 1cnd 9603 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  ( sqr `  ( 1  -  ( x ^
2 ) ) )  =  0 )  -> 
1  e.  CC )
96 sqcl 12187 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x ^ 2 )  e.  CC )
9796adantr 465 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  ( sqr `  ( 1  -  ( x ^
2 ) ) )  =  0 )  -> 
( x ^ 2 )  e.  CC )
9866, 96, 68sylancr 663 . . . . . . . . . . . . 13  |-  ( x  e.  CC  ->  (
1  -  ( x ^ 2 ) )  e.  CC )
9998adantr 465 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  ( sqr `  ( 1  -  ( x ^
2 ) ) )  =  0 )  -> 
( 1  -  (
x ^ 2 ) )  e.  CC )
100 simpr 461 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  ( sqr `  ( 1  -  ( x ^
2 ) ) )  =  0 )  -> 
( sqr `  (
1  -  ( x ^ 2 ) ) )  =  0 )
10199, 100sqr00d 13223 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  ( sqr `  ( 1  -  ( x ^
2 ) ) )  =  0 )  -> 
( 1  -  (
x ^ 2 ) )  =  0 )
10295, 97, 101subeq0d 9929 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  ( sqr `  ( 1  -  ( x ^
2 ) ) )  =  0 )  -> 
1  =  ( x ^ 2 ) )
10394, 102syl5req 2516 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  ( sqr `  ( 1  -  ( x ^
2 ) ) )  =  0 )  -> 
( x ^ 2 )  =  ( 1 ^ 2 ) )
104103ex 434 . . . . . . . 8  |-  ( x  e.  CC  ->  (
( sqr `  (
1  -  ( x ^ 2 ) ) )  =  0  -> 
( x ^ 2 )  =  ( 1 ^ 2 ) ) )
105 sqeqor 12239 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  1  e.  CC )  ->  ( ( x ^
2 )  =  ( 1 ^ 2 )  <-> 
( x  =  1  \/  x  =  -u
1 ) ) )
10666, 105mpan2 671 . . . . . . . 8  |-  ( x  e.  CC  ->  (
( x ^ 2 )  =  ( 1 ^ 2 )  <->  ( x  =  1  \/  x  =  -u 1 ) ) )
107104, 106sylibd 214 . . . . . . 7  |-  ( x  e.  CC  ->  (
( sqr `  (
1  -  ( x ^ 2 ) ) )  =  0  -> 
( x  =  1  \/  x  =  -u
1 ) ) )
108107necon3bd 2674 . . . . . 6  |-  ( x  e.  CC  ->  ( -.  ( x  =  1  \/  x  =  -u
1 )  ->  ( sqr `  ( 1  -  ( x ^ 2 ) ) )  =/=  0 ) )
10950, 93, 108sylc 60 . . . . 5  |-  ( x  e.  D  ->  ( sqr `  ( 1  -  ( x ^ 2 ) ) )  =/=  0 )
11065, 70, 109divnegd 10324 . . . 4  |-  ( x  e.  D  ->  -u (
1  /  ( sqr `  ( 1  -  (
x ^ 2 ) ) ) )  =  ( -u 1  / 
( sqr `  (
1  -  ( x ^ 2 ) ) ) ) )
11164, 110syl5eqr 2517 . . 3  |-  ( x  e.  D  ->  (
0  -  ( 1  /  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) )  =  ( -u 1  / 
( sqr `  (
1  -  ( x ^ 2 ) ) ) ) )
112111mpteq2ia 4524 . 2  |-  ( x  e.  D  |->  ( 0  -  ( 1  / 
( sqr `  (
1  -  ( x ^ 2 ) ) ) ) ) )  =  ( x  e.  D  |->  ( -u 1  /  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) )
1139, 63, 1123eqtri 2495 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 184    \/ wo 368    /\ wa 369    = wceq 1374   T. wtru 1375    e. wcel 1762    =/= wne 2657   _Vcvv 3108    \ cdif 3468    u. cun 3469    C_ wss 3471   {cpr 4024   class class class wbr 4442    |-> cmpt 4500   ran crn 4995    |` cres 4996   -->wf 5577   ` cfv 5581  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484   +oocpnf 9616   -oocmnf 9617   RR*cxr 9618    < clt 9619    - cmin 9796   -ucneg 9797    / cdiv 10197   2c2 10576   (,)cioo 11520   (,]cioc 11521   [,)cico 11522   ^cexp 12124   sqrcsqr 13018   picpi 13655   ↾t crest 14667   TopOpenctopn 14668   topGenctg 14684  ℂfldccnfld 18186  TopOnctopon 19157   Clsdccld 19278    _D cdv 21997  arcsincasin 22916  arccoscacos 22917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ioo 11524  df-ioc 11525  df-ico 11526  df-icc 11527  df-fz 11664  df-fzo 11784  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-fac 12311  df-bc 12338  df-hash 12363  df-shft 12852  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-sum 13460  df-ef 13656  df-sin 13658  df-cos 13659  df-tan 13660  df-pi 13661  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-gsum 14689  df-topgen 14690  df-pt 14691  df-prds 14694  df-xrs 14748  df-qtop 14753  df-imas 14754  df-xps 14756  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-submnd 15773  df-mulg 15856  df-cntz 16145  df-cmn 16591  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-fbas 18182  df-fg 18183  df-cnfld 18187  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-lp 19398  df-perf 19399  df-cn 19489  df-cnp 19490  df-haus 19577  df-cmp 19648  df-tx 19793  df-hmeo 19986  df-fil 20077  df-fm 20169  df-flim 20170  df-flf 20171  df-xms 20553  df-ms 20554  df-tms 20555  df-cncf 21112  df-limc 22000  df-dv 22001  df-log 22667  df-cxp 22668  df-asin 22919  df-acos 22920
This theorem is referenced by:  dvreacos  29672
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