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Theorem tanabsge 22660
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 11559 . . 3  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  ->  A  e.  RR )
2 0re 9596 . . 3  |-  0  e.  RR
3 lttri4 9669 . . 3  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  <  0  \/  A  =  0  \/  0  <  A ) )
41, 2, 3sylancl 662 . 2  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  -> 
( A  <  0  \/  A  =  0  \/  0  <  A ) )
51adantr 465 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  A  e.  RR )
65renegcld 9986 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u A  e.  RR )
71lt0neg1d 10122 . . . . . . . . . . 11  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  -> 
( A  <  0  <->  0  <  -u A ) )
87biimpa 484 . . . . . . . . . 10  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  0  <  -u A )
9 eliooord 11584 . . . . . . . . . . . . 13  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  -> 
( -u ( pi  / 
2 )  <  A  /\  A  <  ( pi 
/  2 ) ) )
109simpld 459 . . . . . . . . . . . 12  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  ->  -u ( pi  /  2
)  <  A )
1110adantr 465 . . . . . . . . . . 11  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u ( pi 
/  2 )  < 
A )
12 halfpire 22618 . . . . . . . . . . . 12  |-  ( pi 
/  2 )  e.  RR
13 ltnegcon1 10053 . . . . . . . . . . . 12  |-  ( ( ( pi  /  2
)  e.  RR  /\  A  e.  RR )  ->  ( -u ( pi 
/  2 )  < 
A  <->  -u A  <  (
pi  /  2 ) ) )
1412, 5, 13sylancr 663 . . . . . . . . . . 11  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( -u (
pi  /  2 )  <  A  <->  -u A  < 
( pi  /  2
) ) )
1511, 14mpbid 210 . . . . . . . . . 10  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u A  < 
( pi  /  2
) )
16 0xr 9640 . . . . . . . . . . 11  |-  0  e.  RR*
1712rexri 9646 . . . . . . . . . . 11  |-  ( pi 
/  2 )  e. 
RR*
18 elioo2 11570 . . . . . . . . . . 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 672 . . . . . . . . . 10  |-  ( -u A  e.  ( 0 (,) ( pi  / 
2 ) )  <->  ( -u A  e.  RR  /\  0  <  -u A  /\  -u A  <  ( pi  /  2
) ) )
206, 8, 15, 19syl3anbrc 1180 . . . . . . . . 9  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u A  e.  ( 0 (,) (
pi  /  2 ) ) )
21 sincosq1sgn 22652 . . . . . . . . 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 463 . . . . . . 7  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  0  <  ( cos `  -u A
) )
2423gt0ne0d 10117 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( cos `  -u A )  =/=  0
)
256, 24retancld 13741 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( tan `  -u A )  e.  RR )
26 tangtx 22659 . . . . . 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 9732 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u A  <_ 
( tan `  -u A
) )
29 ltle 9673 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  <  0  ->  A  <_  0 ) )
301, 2, 29sylancl 662 . . . . . 6  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  -> 
( A  <  0  ->  A  <_  0 ) )
3130imp 429 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  A  <_  0 )
325, 31absnidd 13208 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( abs `  A )  =  -u A )
331recnd 9622 . . . . . . . . . 10  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  ->  A  e.  CC )
3433adantr 465 . . . . . . . . 9  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  A  e.  CC )
3534negnegd 9921 . . . . . . . 8  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u -u A  =  A )
3635fveq2d 5870 . . . . . . 7  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( tan `  -u -u A )  =  ( tan `  A
) )
3734negcld 9917 . . . . . . . 8  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  -u A  e.  CC )
38 tanneg 13744 . . . . . . . 8  |-  ( (
-u A  e.  CC  /\  ( cos `  -u A
)  =/=  0 )  ->  ( tan `  -u -u A
)  =  -u ( tan `  -u A ) )
3937, 24, 38syl2anc 661 . . . . . . 7  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( tan `  -u -u A )  = 
-u ( tan `  -u A
) )
4036, 39eqtr3d 2510 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( tan `  A )  =  -u ( tan `  -u A
) )
4140fveq2d 5870 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( abs `  ( tan `  A
) )  =  ( abs `  -u ( tan `  -u A ) ) )
4225recnd 9622 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( tan `  -u A )  e.  CC )
4342absnegd 13243 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( abs `  -u ( tan `  -u A
) )  =  ( abs `  ( tan `  -u A ) ) )
44 0red 9597 . . . . . . 7  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  0  e.  RR )
45 ltle 9673 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  -u A  e.  RR )  ->  ( 0  <  -u A  ->  0  <_  -u A ) )
462, 6, 45sylancr 663 . . . . . . . 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 9738 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  0  <_  ( tan `  -u A
) )
4925, 48absidd 13217 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( abs `  ( tan `  -u A
) )  =  ( tan `  -u A
) )
5041, 43, 493eqtrd 2512 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( abs `  ( tan `  A
) )  =  ( tan `  -u A
) )
5128, 32, 503brtr4d 4477 . . 3  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  <  0
)  ->  ( abs `  A )  <_  ( abs `  ( tan `  A
) ) )
52 abs0 13081 . . . . . . 7  |-  ( abs `  0 )  =  0
5352, 2eqeltri 2551 . . . . . 6  |-  ( abs `  0 )  e.  RR
5453leidi 10087 . . . . 5  |-  ( abs `  0 )  <_ 
( abs `  0
)
5554a1i 11 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  =  0 )  ->  ( abs `  0 )  <_  ( abs `  0 ) )
56 simpr 461 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  =  0 )  ->  A  = 
0 )
5756fveq2d 5870 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  =  0 )  ->  ( abs `  A )  =  ( abs `  0 ) )
5856fveq2d 5870 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  =  0 )  ->  ( tan `  A )  =  ( tan `  0 ) )
59 tan0 13747 . . . . . 6  |-  ( tan `  0 )  =  0
6058, 59syl6eq 2524 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  =  0 )  ->  ( tan `  A )  =  0 )
6160fveq2d 5870 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  =  0 )  ->  ( abs `  ( tan `  A
) )  =  ( abs `  0 ) )
6255, 57, 613brtr4d 4477 . . 3  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  A  =  0 )  ->  ( abs `  A )  <_  ( abs `  ( tan `  A
) ) )
631adantr 465 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  A  e.  RR )
64 simpr 461 . . . . . . . . . 10  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  0  <  A )
659simprd 463 . . . . . . . . . . 11  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  ->  A  <  ( pi  / 
2 ) )
6665adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  A  <  ( pi  /  2 ) )
67 elioo2 11570 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\  (
pi  /  2 )  e.  RR* )  ->  ( A  e.  ( 0 (,) ( pi  / 
2 ) )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  ( pi  /  2 ) ) ) )
6816, 17, 67mp2an 672 . . . . . . . . . 10  |-  ( A  e.  ( 0 (,) ( pi  /  2
) )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  ( pi  /  2 ) ) )
6963, 64, 66, 68syl3anbrc 1180 . . . . . . . . 9  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  A  e.  ( 0 (,) (
pi  /  2 ) ) )
70 sincosq1sgn 22652 . . . . . . . . 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 463 . . . . . . 7  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  0  <  ( cos `  A ) )
7372gt0ne0d 10117 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  ( cos `  A )  =/=  0
)
7463, 73retancld 13741 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  ( tan `  A )  e.  RR )
75 tangtx 22659 . . . . . 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 9732 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  A  <_  ( tan `  A ) )
78 ltle 9673 . . . . . . 7  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  0  <_  A )
)
792, 1, 78sylancr 663 . . . . . 6  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  -> 
( 0  <  A  ->  0  <_  A )
)
8079imp 429 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  0  <_  A )
8163, 80absidd 13217 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  ( abs `  A )  =  A )
82 0red 9597 . . . . . 6  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  0  e.  RR )
8382, 63, 74, 80, 77letrd 9738 . . . . 5  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  0  <_  ( tan `  A ) )
8474, 83absidd 13217 . . . 4  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  ( abs `  ( tan `  A
) )  =  ( tan `  A ) )
8577, 81, 843brtr4d 4477 . . 3  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  0  <  A
)  ->  ( abs `  A )  <_  ( abs `  ( tan `  A
) ) )
8651, 62, 853jaodan 1294 . 2  |-  ( ( A  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  /\  ( A  <  0  \/  A  =  0  \/  0  < 
A ) )  -> 
( abs `  A
)  <_  ( abs `  ( tan `  A
) ) )
874, 86mpdan 668 1  |-  ( A  e.  ( -u (
pi  /  2 ) (,) ( pi  / 
2 ) )  -> 
( abs `  A
)  <_  ( abs `  ( tan `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492   RR*cxr 9627    < clt 9628    <_ cle 9629   -ucneg 9806    / cdiv 10206   2c2 10585   (,)cioo 11529   abscabs 13030   sincsin 13661   cosccos 13662   tanctan 13663   picpi 13664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-sum 13472  df-ef 13665  df-sin 13667  df-cos 13668  df-tan 13669  df-pi 13670  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-haus 19610  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-xms 20586  df-ms 20587  df-tms 20588  df-cncf 21145  df-limc 22033  df-dv 22034
This theorem is referenced by:  logcnlem4  22782
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