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Theorem atanbndlem 20274
Description: Lemma for atanbnd 20275. (Contributed by Mario Carneiro, 5-Apr-2015.)
Assertion
Ref Expression
atanbndlem  |-  ( A  e.  RR+  ->  (arctan `  A )  e.  (
-u ( pi  / 
2 ) (,) (
pi  /  2 ) ) )

Proof of Theorem atanbndlem
StepHypRef Expression
1 rpre 10407 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR )
2 atanrecl 20260 . . 3  |-  ( A  e.  RR  ->  (arctan `  A )  e.  RR )
31, 2syl 15 . 2  |-  ( A  e.  RR+  ->  (arctan `  A )  e.  RR )
4 pire 19885 . . . . 5  |-  pi  e.  RR
54recni 8894 . . . 4  |-  pi  e.  CC
6 2cn 9861 . . . 4  |-  2  e.  CC
7 2ne0 9874 . . . 4  |-  2  =/=  0
8 divneg 9500 . . . 4  |-  ( ( pi  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
pi  /  2 )  =  ( -u pi  /  2 ) )
95, 6, 7, 8mp3an 1277 . . 3  |-  -u (
pi  /  2 )  =  ( -u pi  /  2 )
10 ax-1cn 8840 . . . . . . . . . . . 12  |-  1  e.  CC
11 ax-icn 8841 . . . . . . . . . . . . 13  |-  _i  e.  CC
121recnd 8906 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  A  e.  CC )
13 mulcl 8866 . . . . . . . . . . . . 13  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
1411, 12, 13sylancr 644 . . . . . . . . . . . 12  |-  ( A  e.  RR+  ->  ( _i  x.  A )  e.  CC )
15 addcl 8864 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
1610, 14, 15sylancr 644 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
17 atanre 20234 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  A  e.  dom arctan )
181, 17syl 15 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  A  e. 
dom arctan )
19 atandm2 20226 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
2018, 19sylib 188 . . . . . . . . . . . 12  |-  ( A  e.  RR+  ->  ( A  e.  CC  /\  (
1  -  ( _i  x.  A ) )  =/=  0  /\  (
1  +  ( _i  x.  A ) )  =/=  0 ) )
2120simp3d 969 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
22 logcl 19979 . . . . . . . . . . 11  |-  ( ( ( 1  +  ( _i  x.  A ) )  e.  CC  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 )  ->  ( log `  (
1  +  ( _i  x.  A ) ) )  e.  CC )
2316, 21, 22syl2anc 642 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
24 subcl 9096 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
2510, 14, 24sylancr 644 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
2620simp2d 968 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
27 logcl 19979 . . . . . . . . . . 11  |-  ( ( ( 1  -  (
_i  x.  A )
)  e.  CC  /\  ( 1  -  (
_i  x.  A )
)  =/=  0 )  ->  ( log `  (
1  -  ( _i  x.  A ) ) )  e.  CC )
2825, 26, 27syl2anc 642 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
2923, 28subcld 9202 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) )  e.  CC )
30 imre 11640 . . . . . . . . 9  |-  ( ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) )  e.  CC  ->  ( Im `  ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( Re `  ( -u _i  x.  (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) ) )
3129, 30syl 15 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( Im
`  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( Re
`  ( -u _i  x.  ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) ) )
32 atanval 20233 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
3318, 32syl 15 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
3433oveq2d 5916 . . . . . . . . . . . 12  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( 2  x.  ( ( _i  /  2 )  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) ) ) )
3511, 6, 7divcan2i 9548 . . . . . . . . . . . . . 14  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
3635oveq1i 5910 . . . . . . . . . . . . 13  |-  ( ( 2  x.  ( _i 
/  2 ) )  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )  =  ( _i  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )
37 2re 9860 . . . . . . . . . . . . . . . 16  |-  2  e.  RR
3837a1i 10 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR+  ->  2  e.  RR )
3938recnd 8906 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  2  e.  CC )
40 halfcl 9984 . . . . . . . . . . . . . . . 16  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
4111, 40ax-mp 8 . . . . . . . . . . . . . . 15  |-  ( _i 
/  2 )  e.  CC
4241a1i 10 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( _i 
/  2 )  e.  CC )
4328, 23subcld 9202 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
4439, 42, 43mulassd 8903 . . . . . . . . . . . . 13  |-  ( A  e.  RR+  ->  ( ( 2  x.  ( _i 
/  2 ) )  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )  =  ( 2  x.  ( ( _i 
/  2 )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) ) )
4536, 44syl5eqr 2362 . . . . . . . . . . . 12  |-  ( A  e.  RR+  ->  ( _i  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )  =  ( 2  x.  ( ( _i 
/  2 )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) ) )
4634, 45eqtr4d 2351 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( _i  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
4723, 28negsubdi2d 9218 . . . . . . . . . . . 12  |-  ( A  e.  RR+  ->  -u (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) )  =  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )
4847oveq2d 5916 . . . . . . . . . . 11  |-  ( A  e.  RR+  ->  ( _i  x.  -u ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( _i  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) ) )
4946, 48eqtr4d 2351 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( _i  x.  -u (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
50 mulneg12 9263 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) )  e.  CC )  ->  ( -u _i  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )  =  ( _i  x.  -u (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
5111, 29, 50sylancr 644 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( -u _i  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( _i  x.  -u ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) ) )
5249, 51eqtr4d 2351 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  (
-u _i  x.  (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
5352fveq2d 5567 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( Re
`  ( 2  x.  (arctan `  A )
) )  =  ( Re `  ( -u _i  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) ) ) )
54 remulcl 8867 . . . . . . . . . 10  |-  ( ( 2  e.  RR  /\  (arctan `  A )  e.  RR )  ->  (
2  x.  (arctan `  A ) )  e.  RR )
5537, 3, 54sylancr 644 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  e.  RR )
5655rered 11756 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( Re
`  ( 2  x.  (arctan `  A )
) )  =  ( 2  x.  (arctan `  A ) ) )
5731, 53, 563eqtr2rd 2355 . . . . . . 7  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  =  ( Im `  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) ) )
58 rpgt0 10412 . . . . . . . . 9  |-  ( A  e.  RR+  ->  0  < 
A )
591rered 11756 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( Re
`  A )  =  A )
6058, 59breqtrrd 4086 . . . . . . . 8  |-  ( A  e.  RR+  ->  0  < 
( Re `  A
) )
61 atanlogsublem 20264 . . . . . . . 8  |-  ( ( A  e.  dom arctan  /\  0  <  ( Re `  A
) )  ->  (
Im `  ( ( log `  ( 1  +  ( _i  x.  A
) ) )  -  ( log `  ( 1  -  ( _i  x.  A ) ) ) ) )  e.  (
-u pi (,) pi ) )
6218, 60, 61syl2anc 642 . . . . . . 7  |-  ( A  e.  RR+  ->  ( Im
`  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  e.  ( -u pi (,) pi ) )
6357, 62eqeltrd 2390 . . . . . 6  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  e.  (
-u pi (,) pi ) )
64 eliooord 10757 . . . . . 6  |-  ( ( 2  x.  (arctan `  A ) )  e.  ( -u pi (,) pi )  ->  ( -u pi  <  ( 2  x.  (arctan `  A )
)  /\  ( 2  x.  (arctan `  A
) )  <  pi ) )
6563, 64syl 15 . . . . 5  |-  ( A  e.  RR+  ->  ( -u pi  <  ( 2  x.  (arctan `  A )
)  /\  ( 2  x.  (arctan `  A
) )  <  pi ) )
6665simpld 445 . . . 4  |-  ( A  e.  RR+  ->  -u pi  <  ( 2  x.  (arctan `  A ) ) )
674renegcli 9153 . . . . . 6  |-  -u pi  e.  RR
6867a1i 10 . . . . 5  |-  ( A  e.  RR+  ->  -u pi  e.  RR )
69 2pos 9873 . . . . . 6  |-  0  <  2
7069a1i 10 . . . . 5  |-  ( A  e.  RR+  ->  0  <  2 )
71 ltdivmul 9673 . . . . 5  |-  ( (
-u pi  e.  RR  /\  (arctan `  A )  e.  RR  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
( ( -u pi  /  2 )  <  (arctan `  A )  <->  -u pi  <  ( 2  x.  (arctan `  A ) ) ) )
7268, 3, 38, 70, 71syl112anc 1186 . . . 4  |-  ( A  e.  RR+  ->  ( (
-u pi  /  2
)  <  (arctan `  A
)  <->  -u pi  <  (
2  x.  (arctan `  A ) ) ) )
7366, 72mpbird 223 . . 3  |-  ( A  e.  RR+  ->  ( -u pi  /  2 )  < 
(arctan `  A )
)
749, 73syl5eqbr 4093 . 2  |-  ( A  e.  RR+  ->  -u (
pi  /  2 )  <  (arctan `  A
) )
7565simprd 449 . . 3  |-  ( A  e.  RR+  ->  ( 2  x.  (arctan `  A
) )  <  pi )
764a1i 10 . . . 4  |-  ( A  e.  RR+  ->  pi  e.  RR )
77 ltmuldiv2 9672 . . . 4  |-  ( ( (arctan `  A )  e.  RR  /\  pi  e.  RR  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
( ( 2  x.  (arctan `  A )
)  <  pi  <->  (arctan `  A
)  <  ( pi  /  2 ) ) )
783, 76, 38, 70, 77syl112anc 1186 . . 3  |-  ( A  e.  RR+  ->  ( ( 2  x.  (arctan `  A ) )  < 
pi 
<->  (arctan `  A )  <  ( pi  /  2
) ) )
7975, 78mpbid 201 . 2  |-  ( A  e.  RR+  ->  (arctan `  A )  <  (
pi  /  2 ) )
80 ressxr 8921 . . . 4  |-  RR  C_  RR*
81 rehalfcl 9985 . . . . . 6  |-  ( pi  e.  RR  ->  (
pi  /  2 )  e.  RR )
824, 81ax-mp 8 . . . . 5  |-  ( pi 
/  2 )  e.  RR
8382renegcli 9153 . . . 4  |-  -u (
pi  /  2 )  e.  RR
8480, 83sselii 3211 . . 3  |-  -u (
pi  /  2 )  e.  RR*
8580, 82sselii 3211 . . 3  |-  ( pi 
/  2 )  e. 
RR*
86 elioo2 10744 . . 3  |-  ( (
-u ( pi  / 
2 )  e.  RR*  /\  ( pi  /  2
)  e.  RR* )  ->  ( (arctan `  A
)  e.  ( -u ( pi  /  2
) (,) ( pi 
/  2 ) )  <-> 
( (arctan `  A
)  e.  RR  /\  -u ( pi  /  2
)  <  (arctan `  A
)  /\  (arctan `  A
)  <  ( pi  /  2 ) ) ) )
8784, 85, 86mp2an 653 . 2  |-  ( (arctan `  A )  e.  (
-u ( pi  / 
2 ) (,) (
pi  /  2 ) )  <->  ( (arctan `  A )  e.  RR  /\  -u ( pi  /  2
)  <  (arctan `  A
)  /\  (arctan `  A
)  <  ( pi  /  2 ) ) )
883, 74, 79, 87syl3anbrc 1136 1  |-  ( A  e.  RR+  ->  (arctan `  A )  e.  (
-u ( pi  / 
2 ) (,) (
pi  /  2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   class class class wbr 4060   dom cdm 4726   ` cfv 5292  (class class class)co 5900   CCcc 8780   RRcr 8781   0cc0 8782   1c1 8783   _ici 8784    + caddc 8785    x. cmul 8787   RR*cxr 8911    < clt 8912    - cmin 9082   -ucneg 9083    / cdiv 9468   2c2 9840   RR+crp 10401   (,)cioo 10703   Recre 11629   Imcim 11630   picpi 12395   logclog 19965  arctancatan 20213
This theorem is referenced by:  atanbnd  20275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-addf 8861  ax-mulf 8862
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-map 6817  df-pm 6818  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-fi 7210  df-sup 7239  df-oi 7270  df-card 7617  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-ioo 10707  df-ioc 10708  df-ico 10709  df-icc 10710  df-fz 10830  df-fzo 10918  df-fl 10972  df-mod 11021  df-seq 11094  df-exp 11152  df-fac 11336  df-bc 11363  df-hash 11385  df-shft 11609  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-limsup 11992  df-clim 12009  df-rlim 12010  df-sum 12206  df-ef 12396  df-sin 12398  df-cos 12399  df-tan 12400  df-pi 12401  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-starv 13270  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-unif 13278  df-hom 13279  df-cco 13280  df-rest 13376  df-topn 13377  df-topgen 13393  df-pt 13394  df-prds 13397  df-xrs 13452  df-0g 13453  df-gsum 13454  df-qtop 13459  df-imas 13460  df-xps 13462  df-mre 13537  df-mrc 13538  df-acs 13540  df-mnd 14416  df-submnd 14465  df-mulg 14541  df-cntz 14842  df-cmn 15140  df-xmet 16425  df-met 16426  df-bl 16427  df-mopn 16428  df-fbas 16429  df-fg 16430  df-cnfld 16433  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-cld 16812  df-ntr 16813  df-cls 16814  df-nei 16891  df-lp 16924  df-perf 16925  df-cn 17013  df-cnp 17014  df-haus 17099  df-tx 17313  df-hmeo 17502  df-fil 17593  df-fm 17685  df-flim 17686  df-flf 17687  df-xms 17937  df-ms 17938  df-tms 17939  df-cncf 18434  df-limc 19269  df-dv 19270  df-log 19967  df-atan 20216
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