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Theorem fourierdlem43 37840
Description:  K is a real function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
fourierdlem43.1  |-  K  =  ( s  e.  (
-u pi [,] pi )  |->  if ( s  =  0 ,  1 ,  ( s  / 
( 2  x.  ( sin `  ( s  / 
2 ) ) ) ) ) )
Assertion
Ref Expression
fourierdlem43  |-  K :
( -u pi [,] pi )
--> RR

Proof of Theorem fourierdlem43
StepHypRef Expression
1 fourierdlem43.1 . 2  |-  K  =  ( s  e.  (
-u pi [,] pi )  |->  if ( s  =  0 ,  1 ,  ( s  / 
( 2  x.  ( sin `  ( s  / 
2 ) ) ) ) ) )
2 1red 9660 . . 3  |-  ( ( s  e.  ( -u pi [,] pi )  /\  s  =  0 )  ->  1  e.  RR )
3 pire 23405 . . . . . . . 8  |-  pi  e.  RR
43a1i 11 . . . . . . 7  |-  ( s  e.  ( -u pi [,] pi )  ->  pi  e.  RR )
54renegcld 10048 . . . . . 6  |-  ( s  e.  ( -u pi [,] pi )  ->  -u pi  e.  RR )
6 id 23 . . . . . 6  |-  ( s  e.  ( -u pi [,] pi )  ->  s  e.  ( -u pi [,] pi ) )
7 eliccre 37440 . . . . . 6  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR  /\  s  e.  ( -u pi [,] pi ) )  -> 
s  e.  RR )
85, 4, 6, 7syl3anc 1265 . . . . 5  |-  ( s  e.  ( -u pi [,] pi )  ->  s  e.  RR )
98adantr 467 . . . 4  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  s  e.  RR )
10 2re 10681 . . . . . 6  |-  2  e.  RR
1110a1i 11 . . . . 5  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  2  e.  RR )
129rehalfcld 10861 . . . . . 6  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  ( s  /  2 )  e.  RR )
1312resincld 14190 . . . . 5  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  ( sin `  ( s  /  2
) )  e.  RR )
1411, 13remulcld 9673 . . . 4  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  ( 2  x.  ( sin `  (
s  /  2 ) ) )  e.  RR )
15 2cnd 10684 . . . . 5  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  2  e.  CC )
1613recnd 9671 . . . . 5  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  ( sin `  ( s  /  2
) )  e.  CC )
17 2ne0 10704 . . . . . 6  |-  2  =/=  0
1817a1i 11 . . . . 5  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  2  =/=  0 )
19 0xr 9689 . . . . . . . . . 10  |-  0  e.  RR*
2019a1i 11 . . . . . . . . 9  |-  ( ( s  e.  ( -u pi [,] pi )  /\  0  <  s )  -> 
0  e.  RR* )
2110, 3remulcli 9659 . . . . . . . . . . 11  |-  ( 2  x.  pi )  e.  RR
2221rexri 9695 . . . . . . . . . 10  |-  ( 2  x.  pi )  e. 
RR*
2322a1i 11 . . . . . . . . 9  |-  ( ( s  e.  ( -u pi [,] pi )  /\  0  <  s )  -> 
( 2  x.  pi )  e.  RR* )
248adantr 467 . . . . . . . . 9  |-  ( ( s  e.  ( -u pi [,] pi )  /\  0  <  s )  -> 
s  e.  RR )
25 simpr 463 . . . . . . . . 9  |-  ( ( s  e.  ( -u pi [,] pi )  /\  0  <  s )  -> 
0  <  s )
2621a1i 11 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,] pi )  ->  (
2  x.  pi )  e.  RR )
275rexrd 9692 . . . . . . . . . . . 12  |-  ( s  e.  ( -u pi [,] pi )  ->  -u pi  e.  RR* )
284rexrd 9692 . . . . . . . . . . . 12  |-  ( s  e.  ( -u pi [,] pi )  ->  pi  e.  RR* )
29 iccleub 11692 . . . . . . . . . . . 12  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  s  e.  ( -u pi [,] pi ) )  ->  s  <_  pi )
3027, 28, 6, 29syl3anc 1265 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,] pi )  ->  s  <_  pi )
31 pirp 23408 . . . . . . . . . . . . 13  |-  pi  e.  RR+
32 2timesgt 37349 . . . . . . . . . . . . 13  |-  ( pi  e.  RR+  ->  pi  <  ( 2  x.  pi ) )
3331, 32ax-mp 5 . . . . . . . . . . . 12  |-  pi  <  ( 2  x.  pi )
3433a1i 11 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,] pi )  ->  pi  <  ( 2  x.  pi ) )
358, 4, 26, 30, 34lelttrd 9795 . . . . . . . . . 10  |-  ( s  e.  ( -u pi [,] pi )  ->  s  <  ( 2  x.  pi ) )
3635adantr 467 . . . . . . . . 9  |-  ( ( s  e.  ( -u pi [,] pi )  /\  0  <  s )  -> 
s  <  ( 2  x.  pi ) )
3720, 23, 24, 25, 36eliood 37432 . . . . . . . 8  |-  ( ( s  e.  ( -u pi [,] pi )  /\  0  <  s )  -> 
s  e.  ( 0 (,) ( 2  x.  pi ) ) )
38 sinaover2ne0 37569 . . . . . . . 8  |-  ( s  e.  ( 0 (,) ( 2  x.  pi ) )  ->  ( sin `  ( s  / 
2 ) )  =/=  0 )
3937, 38syl 17 . . . . . . 7  |-  ( ( s  e.  ( -u pi [,] pi )  /\  0  <  s )  -> 
( sin `  (
s  /  2 ) )  =/=  0 )
4039adantlr 720 . . . . . 6  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  0  <  s )  ->  ( sin `  ( s  / 
2 ) )  =/=  0 )
418ad2antrr 731 . . . . . . . 8  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  -> 
s  e.  RR )
42 iccgelb 11693 . . . . . . . . . 10  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  s  e.  ( -u pi [,] pi ) )  ->  -u pi  <_  s )
4327, 28, 6, 42syl3anc 1265 . . . . . . . . 9  |-  ( s  e.  ( -u pi [,] pi )  ->  -u pi  <_  s )
4443ad2antrr 731 . . . . . . . 8  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  ->  -u pi  <_  s )
45 0red 9646 . . . . . . . . 9  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  -> 
0  e.  RR )
46 neqne 37241 . . . . . . . . . 10  |-  ( -.  s  =  0  -> 
s  =/=  0 )
4746ad2antlr 732 . . . . . . . . 9  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  -> 
s  =/=  0 )
48 simpr 463 . . . . . . . . 9  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  ->  -.  0  <  s )
4941, 45, 47, 48lttri5d 37365 . . . . . . . 8  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  -> 
s  <  0 )
505ad2antrr 731 . . . . . . . . 9  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  ->  -u pi  e.  RR )
51 elico2 11700 . . . . . . . . 9  |-  ( (
-u pi  e.  RR  /\  0  e.  RR* )  ->  ( s  e.  (
-u pi [,) 0
)  <->  ( s  e.  RR  /\  -u pi  <_  s  /\  s  <  0 ) ) )
5250, 19, 51sylancl 667 . . . . . . . 8  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  -> 
( s  e.  (
-u pi [,) 0
)  <->  ( s  e.  RR  /\  -u pi  <_  s  /\  s  <  0 ) ) )
5341, 44, 49, 52mpbir3and 1189 . . . . . . 7  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  -> 
s  e.  ( -u pi [,) 0 ) )
543renegcli 9937 . . . . . . . . . . . . . . 15  |-  -u pi  e.  RR
55 elicore 11689 . . . . . . . . . . . . . . 15  |-  ( (
-u pi  e.  RR  /\  s  e.  ( -u pi [,) 0 ) )  ->  s  e.  RR )
5654, 55mpan 675 . . . . . . . . . . . . . 14  |-  ( s  e.  ( -u pi [,) 0 )  ->  s  e.  RR )
5756recnd 9671 . . . . . . . . . . . . 13  |-  ( s  e.  ( -u pi [,) 0 )  ->  s  e.  CC )
58 2cnd 10684 . . . . . . . . . . . . 13  |-  ( s  e.  ( -u pi [,) 0 )  ->  2  e.  CC )
5917a1i 11 . . . . . . . . . . . . 13  |-  ( s  e.  ( -u pi [,) 0 )  ->  2  =/=  0 )
6057, 58, 59divnegd 10398 . . . . . . . . . . . 12  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u (
s  /  2 )  =  ( -u s  /  2 ) )
6160eqcomd 2431 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( -u s  /  2 )  =  -u ( s  / 
2 ) )
6261fveq2d 5883 . . . . . . . . . 10  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( sin `  ( -u s  /  2 ) )  =  ( sin `  -u (
s  /  2 ) ) )
6362negeqd 9871 . . . . . . . . 9  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u ( sin `  ( -u s  /  2 ) )  =  -u ( sin `  -u (
s  /  2 ) ) )
6457halfcld 10859 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,) 0 )  ->  (
s  /  2 )  e.  CC )
65 sinneg 14193 . . . . . . . . . . 11  |-  ( ( s  /  2 )  e.  CC  ->  ( sin `  -u ( s  / 
2 ) )  = 
-u ( sin `  (
s  /  2 ) ) )
6664, 65syl 17 . . . . . . . . . 10  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( sin `  -u ( s  / 
2 ) )  = 
-u ( sin `  (
s  /  2 ) ) )
6766negeqd 9871 . . . . . . . . 9  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u ( sin `  -u ( s  / 
2 ) )  = 
-u -u ( sin `  (
s  /  2 ) ) )
6864sincld 14177 . . . . . . . . . 10  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( sin `  ( s  / 
2 ) )  e.  CC )
6968negnegd 9979 . . . . . . . . 9  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u -u ( sin `  ( s  / 
2 ) )  =  ( sin `  (
s  /  2 ) ) )
7063, 67, 693eqtrd 2468 . . . . . . . 8  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u ( sin `  ( -u s  /  2 ) )  =  ( sin `  (
s  /  2 ) ) )
7157negcld 9975 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u s  e.  CC )
7271halfcld 10859 . . . . . . . . . 10  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( -u s  /  2 )  e.  CC )
7372sincld 14177 . . . . . . . . 9  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( sin `  ( -u s  /  2 ) )  e.  CC )
7419a1i 11 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,) 0 )  ->  0  e.  RR* )
7522a1i 11 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,) 0 )  ->  (
2  x.  pi )  e.  RR* )
7656renegcld 10048 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u s  e.  RR )
7754a1i 11 . . . . . . . . . . . . . 14  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u pi  e.  RR )
7877rexrd 9692 . . . . . . . . . . . . 13  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u pi  e.  RR* )
79 id 23 . . . . . . . . . . . . 13  |-  ( s  e.  ( -u pi [,) 0 )  ->  s  e.  ( -u pi [,) 0 ) )
80 icoltub 37444 . . . . . . . . . . . . 13  |-  ( (
-u pi  e.  RR*  /\  0  e.  RR*  /\  s  e.  ( -u pi [,) 0 ) )  -> 
s  <  0 )
8178, 74, 79, 80syl3anc 1265 . . . . . . . . . . . 12  |-  ( s  e.  ( -u pi [,) 0 )  ->  s  <  0 )
8256lt0neg1d 10185 . . . . . . . . . . . 12  |-  ( s  e.  ( -u pi [,) 0 )  ->  (
s  <  0  <->  0  <  -u s ) )
8381, 82mpbid 214 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,) 0 )  ->  0  <  -u s )
843a1i 11 . . . . . . . . . . . 12  |-  ( s  e.  ( -u pi [,) 0 )  ->  pi  e.  RR )
8521a1i 11 . . . . . . . . . . . 12  |-  ( s  e.  ( -u pi [,) 0 )  ->  (
2  x.  pi )  e.  RR )
86 icogelb 11688 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  0  e.  RR*  /\  s  e.  ( -u pi [,) 0 ) )  ->  -u pi  <_  s )
8778, 74, 79, 86syl3anc 1265 . . . . . . . . . . . . 13  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u pi  <_  s )
8884, 56, 87lenegcon1d 10197 . . . . . . . . . . . 12  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u s  <_  pi )
8933a1i 11 . . . . . . . . . . . 12  |-  ( s  e.  ( -u pi [,) 0 )  ->  pi  <  ( 2  x.  pi ) )
9076, 84, 85, 88, 89lelttrd 9795 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u s  <  ( 2  x.  pi ) )
9174, 75, 76, 83, 90eliood 37432 . . . . . . . . . 10  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u s  e.  ( 0 (,) (
2  x.  pi ) ) )
92 sinaover2ne0 37569 . . . . . . . . . 10  |-  ( -u s  e.  ( 0 (,) ( 2  x.  pi ) )  -> 
( sin `  ( -u s  /  2 ) )  =/=  0 )
9391, 92syl 17 . . . . . . . . 9  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( sin `  ( -u s  /  2 ) )  =/=  0 )
9473, 93negne0d 9986 . . . . . . . 8  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u ( sin `  ( -u s  /  2 ) )  =/=  0 )
9570, 94eqnetrrd 2719 . . . . . . 7  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( sin `  ( s  / 
2 ) )  =/=  0 )
9653, 95syl 17 . . . . . 6  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  -> 
( sin `  (
s  /  2 ) )  =/=  0 )
9740, 96pm2.61dan 799 . . . . 5  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  ( sin `  ( s  /  2
) )  =/=  0
)
9815, 16, 18, 97mulne0d 10266 . . . 4  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  ( 2  x.  ( sin `  (
s  /  2 ) ) )  =/=  0
)
999, 14, 98redivcld 10437 . . 3  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  ( s  /  ( 2  x.  ( sin `  (
s  /  2 ) ) ) )  e.  RR )
1002, 99ifclda 3942 . 2  |-  ( s  e.  ( -u pi [,] pi )  ->  if ( s  =  0 ,  1 ,  ( s  /  ( 2  x.  ( sin `  (
s  /  2 ) ) ) ) )  e.  RR )
1011, 100fmpti 6058 1  |-  K :
( -u pi [,] pi )
--> RR
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   ifcif 3910   class class class wbr 4421    |-> cmpt 4480   -->wf 5595   ` cfv 5599  (class class class)co 6303   CCcc 9539   RRcr 9540   0cc0 9541   1c1 9542    x. cmul 9546   RR*cxr 9676    < clt 9677    <_ cle 9678   -ucneg 9863    / cdiv 10271   2c2 10661   RR+crp 11304   (,)cioo 11637   [,)cico 11639   [,]cicc 11640   sincsin 14109   picpi 14112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619  ax-addf 9620  ax-mulf 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-fi 7929  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-q 11267  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-ioo 11641  df-ioc 11642  df-ico 11643  df-icc 11644  df-fz 11787  df-fzo 11918  df-fl 12029  df-mod 12098  df-seq 12215  df-exp 12274  df-fac 12461  df-bc 12489  df-hash 12517  df-shft 13124  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-cld 20026  df-ntr 20027  df-cls 20028  df-nei 20106  df-lp 20144  df-perf 20145  df-cn 20235  df-cnp 20236  df-haus 20323  df-tx 20569  df-hmeo 20762  df-fil 20853  df-fm 20945  df-flim 20946  df-flf 20947  df-xms 21327  df-ms 21328  df-tms 21329  df-cncf 21902  df-limc 22813  df-dv 22814
This theorem is referenced by:  fourierdlem55  37851  fourierdlem62  37858  fourierdlem66  37862  fourierdlem77  37873  fourierdlem85  37881  fourierdlem88  37884  fourierdlem103  37899  fourierdlem104  37900
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