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Theorem tanabsge 20282
Description: The tangent function is greater than or equal to its argument in absolute value. (Contributed by Mario Carneiro, 25-Feb-2015.)
Assertion
Ref Expression
tanabsge  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  -> 
( abs `  A
)  <_  ( abs `  ( tan `  A
) ) )

Proof of Theorem tanabsge
StepHypRef Expression
1 elioore 10879 . . 3  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  ->  A  e.  RR )
2 0re 9025 . . 3  |-  0  e.  RR
3 lttri4 9093 . . 3  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  <  0  \/  A  =  0  \/  0  <  A ) )
41, 2, 3sylancl 644 . 2  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  -> 
( A  <  0  \/  A  =  0  \/  0  <  A ) )
51adantr 452 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  A  e.  RR )
65renegcld 9397 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u A  e.  RR )
71lt0neg1d 9529 . . . . . . . . . . 11  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  -> 
( A  <  0  <->  0  <  -u A ) )
87biimpa 471 . . . . . . . . . 10  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  0  <  -u A )
9 eliooord 10903 . . . . . . . . . . . . 13  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  -> 
( -u ( pi  / 
2 )  <  A  /\  A  <  ( pi 
/  2 ) ) )
109simpld 446 . . . . . . . . . . . 12  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  ->  -u ( pi  /  2
)  <  A )
1110adantr 452 . . . . . . . . . . 11  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u ( pi 
/  2 )  < 
A )
12 halfpire 20243 . . . . . . . . . . . 12  |-  ( pi 
/  2 )  e.  RR
13 ltnegcon1 9462 . . . . . . . . . . . 12  |-  ( ( ( pi  /  2
)  e.  RR  /\  A  e.  RR )  ->  ( -u ( pi 
/  2 )  < 
A  <->  -u A  <  (
pi  /  2 ) ) )
1412, 5, 13sylancr 645 . . . . . . . . . . 11  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( -u (
pi  /  2 )  <  A  <->  -u A  < 
( pi  /  2
) ) )
1511, 14mpbid 202 . . . . . . . . . 10  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u A  < 
( pi  /  2
) )
16 0xr 9065 . . . . . . . . . . 11  |-  0  e.  RR*
1712rexri 9071 . . . . . . . . . . 11  |-  ( pi 
/  2 )  e. 
RR*
18 elioo2 10890 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\  (
pi  /  2 )  e.  RR* )  ->  ( -u A  e.  ( 0 (,) ( pi  / 
2 ) )  <->  ( -u A  e.  RR  /\  0  <  -u A  /\  -u A  <  ( pi  /  2
) ) ) )
1916, 17, 18mp2an 654 . . . . . . . . . 10  |-  ( -u A  e.  ( 0 (,) ( pi  / 
2 ) )  <->  ( -u A  e.  RR  /\  0  <  -u A  /\  -u A  <  ( pi  /  2
) ) )
206, 8, 15, 19syl3anbrc 1138 . . . . . . . . 9  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u A  e.  ( 0 (,) (
pi  /  2 ) ) )
21 sincosq1sgn 20274 . . . . . . . . 9  |-  ( -u A  e.  ( 0 (,) ( pi  / 
2 ) )  -> 
( 0  <  ( sin `  -u A )  /\  0  <  ( cos `  -u A
) ) )
2220, 21syl 16 . . . . . . . 8  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( 0  <  ( sin `  -u A
)  /\  0  <  ( cos `  -u A
) ) )
2322simprd 450 . . . . . . 7  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  0  <  ( cos `  -u A
) )
2423gt0ne0d 9524 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( cos `  -u A )  =/=  0
)
256, 24retancld 12674 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( tan `  -u A )  e.  RR )
26 tangtx 20281 . . . . . 6  |-  ( -u A  e.  ( 0 (,) ( pi  / 
2 ) )  ->  -u A  <  ( tan `  -u A ) )
2720, 26syl 16 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u A  < 
( tan `  -u A
) )
286, 25, 27ltled 9154 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u A  <_ 
( tan `  -u A
) )
29 ltle 9097 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  <  0  ->  A  <_  0 ) )
301, 2, 29sylancl 644 . . . . . 6  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  -> 
( A  <  0  ->  A  <_  0 ) )
3130imp 419 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  A  <_  0 )
325, 31absnidd 12144 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( abs `  A )  =  -u A )
331recnd 9048 . . . . . . . . . 10  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  ->  A  e.  CC )
3433adantr 452 . . . . . . . . 9  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  A  e.  CC )
3534negnegd 9335 . . . . . . . 8  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u -u A  =  A )
3635fveq2d 5673 . . . . . . 7  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( tan `  -u -u A )  =  ( tan `  A
) )
3734negcld 9331 . . . . . . . 8  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u A  e.  CC )
38 tanneg 12677 . . . . . . . 8  |-  ( (
-u A  e.  CC  /\  ( cos `  -u A
)  =/=  0 )  ->  ( tan `  -u -u A
)  =  -u ( tan `  -u A ) )
3937, 24, 38syl2anc 643 . . . . . . 7  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( tan `  -u -u A )  = 
-u ( tan `  -u A
) )
4036, 39eqtr3d 2422 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( tan `  A )  =  -u ( tan `  -u A
) )
4140fveq2d 5673 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( abs `  ( tan `  A
) )  =  ( abs `  -u ( tan `  -u A ) ) )
4225recnd 9048 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( tan `  -u A )  e.  CC )
4342absnegd 12179 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( abs `  -u ( tan `  -u A
) )  =  ( abs `  ( tan `  -u A ) ) )
442a1i 11 . . . . . . 7  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  0  e.  RR )
45 ltle 9097 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  -u A  e.  RR )  ->  ( 0  <  -u A  ->  0  <_  -u A ) )
462, 6, 45sylancr 645 . . . . . . . 8  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( 0  <  -u A  ->  0  <_ 
-u A ) )
478, 46mpd 15 . . . . . . 7  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  0  <_  -u A )
4844, 6, 25, 47, 28letrd 9160 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  0  <_  ( tan `  -u A
) )
4925, 48absidd 12153 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( abs `  ( tan `  -u A
) )  =  ( tan `  -u A
) )
5041, 43, 493eqtrd 2424 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( abs `  ( tan `  A
) )  =  ( tan `  -u A
) )
5128, 32, 503brtr4d 4184 . . 3  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( abs `  A )  <_  ( abs `  ( tan `  A
) ) )
52 abs0 12018 . . . . . . 7  |-  ( abs `  0 )  =  0
5352, 2eqeltri 2458 . . . . . 6  |-  ( abs `  0 )  e.  RR
5453leidi 9494 . . . . 5  |-  ( abs `  0 )  <_ 
( abs `  0
)
5554a1i 11 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  =  0 )  ->  ( abs `  0 )  <_  ( abs `  0 ) )
56 simpr 448 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  =  0 )  ->  A  = 
0 )
5756fveq2d 5673 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  =  0 )  ->  ( abs `  A )  =  ( abs `  0 ) )
5856fveq2d 5673 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  =  0 )  ->  ( tan `  A )  =  ( tan `  0 ) )
59 tan0 12680 . . . . . 6  |-  ( tan `  0 )  =  0
6058, 59syl6eq 2436 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  =  0 )  ->  ( tan `  A )  =  0 )
6160fveq2d 5673 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  =  0 )  ->  ( abs `  ( tan `  A
) )  =  ( abs `  0 ) )
6255, 57, 613brtr4d 4184 . . 3  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  =  0 )  ->  ( abs `  A )  <_  ( abs `  ( tan `  A
) ) )
631adantr 452 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  A  e.  RR )
64 simpr 448 . . . . . . . . . 10  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  0  <  A )
659simprd 450 . . . . . . . . . . 11  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  ->  A  <  ( pi  / 
2 ) )
6665adantr 452 . . . . . . . . . 10  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  A  <  ( pi  /  2 ) )
67 elioo2 10890 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\  (
pi  /  2 )  e.  RR* )  ->  ( A  e.  ( 0 (,) ( pi  / 
2 ) )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  ( pi  /  2 ) ) ) )
6816, 17, 67mp2an 654 . . . . . . . . . 10  |-  ( A  e.  ( 0 (,) ( pi  /  2
) )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  ( pi  /  2 ) ) )
6963, 64, 66, 68syl3anbrc 1138 . . . . . . . . 9  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  A  e.  ( 0 (,) (
pi  /  2 ) ) )
70 sincosq1sgn 20274 . . . . . . . . 9  |-  ( A  e.  ( 0 (,) ( pi  /  2
) )  ->  (
0  <  ( sin `  A )  /\  0  <  ( cos `  A
) ) )
7169, 70syl 16 . . . . . . . 8  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  ( 0  <  ( sin `  A
)  /\  0  <  ( cos `  A ) ) )
7271simprd 450 . . . . . . 7  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  0  <  ( cos `  A ) )
7372gt0ne0d 9524 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  ( cos `  A )  =/=  0
)
7463, 73retancld 12674 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  ( tan `  A )  e.  RR )
75 tangtx 20281 . . . . . 6  |-  ( A  e.  ( 0 (,) ( pi  /  2
) )  ->  A  <  ( tan `  A
) )
7669, 75syl 16 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  A  <  ( tan `  A ) )
7763, 74, 76ltled 9154 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  A  <_  ( tan `  A ) )
78 ltle 9097 . . . . . . 7  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  0  <_  A )
)
792, 1, 78sylancr 645 . . . . . 6  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  -> 
( 0  <  A  ->  0  <_  A )
)
8079imp 419 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  0  <_  A )
8163, 80absidd 12153 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  ( abs `  A )  =  A )
822a1i 11 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  0  e.  RR )
8382, 63, 74, 80, 77letrd 9160 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  0  <_  ( tan `  A ) )
8474, 83absidd 12153 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  ( abs `  ( tan `  A
) )  =  ( tan `  A ) )
8577, 81, 843brtr4d 4184 . . 3  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  ( abs `  A )  <_  ( abs `  ( tan `  A
) ) )
8651, 62, 853jaodan 1250 . 2  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  ( A  <  0  \/  A  =  0  \/  0  < 
A ) )  -> 
( abs `  A
)  <_  ( abs `  ( tan `  A
) ) )
874, 86mpdan 650 1  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  -> 
( abs `  A
)  <_  ( abs `  ( tan `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   CCcc 8922   RRcr 8923   0cc0 8924   RR*cxr 9053    < clt 9054    <_ cle 9055   -ucneg 9225    / cdiv 9610   2c2 9982   (,)cioo 10849   abscabs 11967   sincsin 12594   cosccos 12595   tanctan 12596   picpi 12597
This theorem is referenced by:  logcnlem4  20404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ioc 10854  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-seq 11252  df-exp 11311  df-fac 11495  df-bc 11522  df-hash 11547  df-shft 11810  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-limsup 12193  df-clim 12210  df-rlim 12211  df-sum 12408  df-ef 12598  df-sin 12600  df-cos 12601  df-tan 12602  df-pi 12603  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667  df-mulg 14743  df-cntz 15044  df-cmn 15342  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-lp 17124  df-perf 17125  df-cn 17214  df-cnp 17215  df-haus 17302  df-tx 17516  df-hmeo 17709  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-tms 18262  df-cncf 18780  df-limc 19621  df-dv 19622
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