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Theorem fourierdlem43 32135
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 9628 . . 3  |-  ( ( s  e.  ( -u pi [,] pi )  /\  s  =  0 )  ->  1  e.  RR )
3 pire 22977 . . . . . . . 8  |-  pi  e.  RR
43a1i 11 . . . . . . 7  |-  ( s  e.  ( -u pi [,] pi )  ->  pi  e.  RR )
54renegcld 10007 . . . . . 6  |-  ( s  e.  ( -u pi [,] pi )  ->  -u pi  e.  RR )
6 id 22 . . . . . 6  |-  ( s  e.  ( -u pi [,] pi )  ->  s  e.  ( -u pi [,] pi ) )
7 eliccre 31743 . . . . . 6  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR  /\  s  e.  ( -u pi [,] pi ) )  -> 
s  e.  RR )
85, 4, 6, 7syl3anc 1228 . . . . 5  |-  ( s  e.  ( -u pi [,] pi )  ->  s  e.  RR )
98adantr 465 . . . 4  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  s  e.  RR )
10 2re 10626 . . . . . 6  |-  2  e.  RR
1110a1i 11 . . . . 5  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  2  e.  RR )
129rehalfcld 10806 . . . . . 6  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  ( s  /  2 )  e.  RR )
1312resincld 13890 . . . . 5  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  ( sin `  ( s  /  2
) )  e.  RR )
1411, 13remulcld 9641 . . . 4  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  ( 2  x.  ( sin `  (
s  /  2 ) ) )  e.  RR )
15 2cnd 10629 . . . . 5  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  2  e.  CC )
1613recnd 9639 . . . . 5  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  ( sin `  ( s  /  2
) )  e.  CC )
17 2ne0 10649 . . . . . 6  |-  2  =/=  0
1817a1i 11 . . . . 5  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  2  =/=  0 )
19 0xr 9657 . . . . . . . . . 10  |-  0  e.  RR*
2019a1i 11 . . . . . . . . 9  |-  ( ( s  e.  ( -u pi [,] pi )  /\  0  <  s )  -> 
0  e.  RR* )
2110, 3remulcli 9627 . . . . . . . . . . 11  |-  ( 2  x.  pi )  e.  RR
2221rexri 9663 . . . . . . . . . 10  |-  ( 2  x.  pi )  e. 
RR*
2322a1i 11 . . . . . . . . 9  |-  ( ( s  e.  ( -u pi [,] pi )  /\  0  <  s )  -> 
( 2  x.  pi )  e.  RR* )
248adantr 465 . . . . . . . . 9  |-  ( ( s  e.  ( -u pi [,] pi )  /\  0  <  s )  -> 
s  e.  RR )
25 simpr 461 . . . . . . . . 9  |-  ( ( s  e.  ( -u pi [,] pi )  /\  0  <  s )  -> 
0  <  s )
2621a1i 11 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,] pi )  ->  (
2  x.  pi )  e.  RR )
275rexrd 9660 . . . . . . . . . . . 12  |-  ( s  e.  ( -u pi [,] pi )  ->  -u pi  e.  RR* )
284rexrd 9660 . . . . . . . . . . . 12  |-  ( s  e.  ( -u pi [,] pi )  ->  pi  e.  RR* )
29 iccleub 11605 . . . . . . . . . . . 12  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  s  e.  ( -u pi [,] pi ) )  ->  s  <_  pi )
3027, 28, 6, 29syl3anc 1228 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,] pi )  ->  s  <_  pi )
31 pirp 22980 . . . . . . . . . . . . 13  |-  pi  e.  RR+
32 2timesgt 31678 . . . . . . . . . . . . 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 9757 . . . . . . . . . 10  |-  ( s  e.  ( -u pi [,] pi )  ->  s  <  ( 2  x.  pi ) )
3635adantr 465 . . . . . . . . 9  |-  ( ( s  e.  ( -u pi [,] pi )  /\  0  <  s )  -> 
s  <  ( 2  x.  pi ) )
3720, 23, 24, 25, 36eliood 31734 . . . . . . . 8  |-  ( ( s  e.  ( -u pi [,] pi )  /\  0  <  s )  -> 
s  e.  ( 0 (,) ( 2  x.  pi ) ) )
38 sinaover2ne0 31871 . . . . . . . 8  |-  ( s  e.  ( 0 (,) ( 2  x.  pi ) )  ->  ( sin `  ( s  / 
2 ) )  =/=  0 )
3937, 38syl 16 . . . . . . 7  |-  ( ( s  e.  ( -u pi [,] pi )  /\  0  <  s )  -> 
( sin `  (
s  /  2 ) )  =/=  0 )
4039adantlr 714 . . . . . 6  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  0  <  s )  ->  ( sin `  ( s  / 
2 ) )  =/=  0 )
418ad2antrr 725 . . . . . . . 8  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  -> 
s  e.  RR )
42 iccgelb 11606 . . . . . . . . . 10  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  s  e.  ( -u pi [,] pi ) )  ->  -u pi  <_  s )
4327, 28, 6, 42syl3anc 1228 . . . . . . . . 9  |-  ( s  e.  ( -u pi [,] pi )  ->  -u pi  <_  s )
4443ad2antrr 725 . . . . . . . 8  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  ->  -u pi  <_  s )
45 0red 9614 . . . . . . . . 9  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  -> 
0  e.  RR )
46 neqne 31637 . . . . . . . . . 10  |-  ( -.  s  =  0  -> 
s  =/=  0 )
4746ad2antlr 726 . . . . . . . . 9  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  -> 
s  =/=  0 )
48 simpr 461 . . . . . . . . 9  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  ->  -.  0  <  s )
4941, 45, 47, 48lttri5d 31702 . . . . . . . 8  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  -> 
s  <  0 )
505ad2antrr 725 . . . . . . . . 9  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  ->  -u pi  e.  RR )
51 elico2 11613 . . . . . . . . 9  |-  ( (
-u pi  e.  RR  /\  0  e.  RR* )  ->  ( s  e.  (
-u pi [,) 0
)  <->  ( s  e.  RR  /\  -u pi  <_  s  /\  s  <  0 ) ) )
5250, 19, 51sylancl 662 . . . . . . . 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 1179 . . . . . . 7  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  -> 
s  e.  ( -u pi [,) 0 ) )
543renegcli 9899 . . . . . . . . . . . . . . 15  |-  -u pi  e.  RR
55 elicore 31740 . . . . . . . . . . . . . . 15  |-  ( (
-u pi  e.  RR  /\  s  e.  ( -u pi [,) 0 ) )  ->  s  e.  RR )
5654, 55mpan 670 . . . . . . . . . . . . . 14  |-  ( s  e.  ( -u pi [,) 0 )  ->  s  e.  RR )
5756recnd 9639 . . . . . . . . . . . . 13  |-  ( s  e.  ( -u pi [,) 0 )  ->  s  e.  CC )
58 2cnd 10629 . . . . . . . . . . . . 13  |-  ( s  e.  ( -u pi [,) 0 )  ->  2  e.  CC )
5917a1i 11 . . . . . . . . . . . . 13  |-  ( s  e.  ( -u pi [,) 0 )  ->  2  =/=  0 )
6057, 58, 59divnegd 10354 . . . . . . . . . . . 12  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u (
s  /  2 )  =  ( -u s  /  2 ) )
6160eqcomd 2465 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( -u s  /  2 )  =  -u ( s  / 
2 ) )
6261fveq2d 5876 . . . . . . . . . 10  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( sin `  ( -u s  /  2 ) )  =  ( sin `  -u (
s  /  2 ) ) )
6362negeqd 9833 . . . . . . . . 9  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u ( sin `  ( -u s  /  2 ) )  =  -u ( sin `  -u (
s  /  2 ) ) )
6457halfcld 10804 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,) 0 )  ->  (
s  /  2 )  e.  CC )
65 sinneg 13893 . . . . . . . . . . 11  |-  ( ( s  /  2 )  e.  CC  ->  ( sin `  -u ( s  / 
2 ) )  = 
-u ( sin `  (
s  /  2 ) ) )
6664, 65syl 16 . . . . . . . . . 10  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( sin `  -u ( s  / 
2 ) )  = 
-u ( sin `  (
s  /  2 ) ) )
6766negeqd 9833 . . . . . . . . 9  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u ( sin `  -u ( s  / 
2 ) )  = 
-u -u ( sin `  (
s  /  2 ) ) )
6864sincld 13877 . . . . . . . . . 10  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( sin `  ( s  / 
2 ) )  e.  CC )
6968negnegd 9941 . . . . . . . . 9  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u -u ( sin `  ( s  / 
2 ) )  =  ( sin `  (
s  /  2 ) ) )
7063, 67, 693eqtrd 2502 . . . . . . . 8  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u ( sin `  ( -u s  /  2 ) )  =  ( sin `  (
s  /  2 ) ) )
7157negcld 9937 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u s  e.  CC )
7271halfcld 10804 . . . . . . . . . 10  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( -u s  /  2 )  e.  CC )
7372sincld 13877 . . . . . . . . 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 10007 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u s  e.  RR )
7754a1i 11 . . . . . . . . . . . . . 14  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u pi  e.  RR )
7877rexrd 9660 . . . . . . . . . . . . 13  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u pi  e.  RR* )
79 id 22 . . . . . . . . . . . . 13  |-  ( s  e.  ( -u pi [,) 0 )  ->  s  e.  ( -u pi [,) 0 ) )
80 icoltub 31748 . . . . . . . . . . . . 13  |-  ( (
-u pi  e.  RR*  /\  0  e.  RR*  /\  s  e.  ( -u pi [,) 0 ) )  -> 
s  <  0 )
8178, 74, 79, 80syl3anc 1228 . . . . . . . . . . . 12  |-  ( s  e.  ( -u pi [,) 0 )  ->  s  <  0 )
8256lt0neg1d 10143 . . . . . . . . . . . 12  |-  ( s  e.  ( -u pi [,) 0 )  ->  (
s  <  0  <->  0  <  -u s ) )
8381, 82mpbid 210 . . . . . . . . . . 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 31745 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  0  e.  RR*  /\  s  e.  ( -u pi [,) 0 ) )  ->  -u pi  <_  s )
8778, 74, 79, 86syl3anc 1228 . . . . . . . . . . . . 13  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u pi  <_  s )
8884, 56, 87lenegcon1d 10155 . . . . . . . . . . . 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 9757 . . . . . . . . . . 11  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u s  <  ( 2  x.  pi ) )
9174, 75, 76, 83, 90eliood 31734 . . . . . . . . . 10  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u s  e.  ( 0 (,) (
2  x.  pi ) ) )
92 sinaover2ne0 31871 . . . . . . . . . 10  |-  ( -u s  e.  ( 0 (,) ( 2  x.  pi ) )  -> 
( sin `  ( -u s  /  2 ) )  =/=  0 )
9391, 92syl 16 . . . . . . . . 9  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( sin `  ( -u s  /  2 ) )  =/=  0 )
9473, 93negne0d 9948 . . . . . . . 8  |-  ( s  e.  ( -u pi [,) 0 )  ->  -u ( sin `  ( -u s  /  2 ) )  =/=  0 )
9570, 94eqnetrrd 2751 . . . . . . 7  |-  ( s  e.  ( -u pi [,) 0 )  ->  ( sin `  ( s  / 
2 ) )  =/=  0 )
9653, 95syl 16 . . . . . 6  |-  ( ( ( s  e.  (
-u pi [,] pi )  /\  -.  s  =  0 )  /\  -.  0  <  s )  -> 
( sin `  (
s  /  2 ) )  =/=  0 )
9740, 96pm2.61dan 791 . . . . 5  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  ( sin `  ( s  /  2
) )  =/=  0
)
9815, 16, 18, 97mulne0d 10222 . . . 4  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  ( 2  x.  ( sin `  (
s  /  2 ) ) )  =/=  0
)
999, 14, 98redivcld 10393 . . 3  |-  ( ( s  e.  ( -u pi [,] pi )  /\  -.  s  =  0
)  ->  ( s  /  ( 2  x.  ( sin `  (
s  /  2 ) ) ) )  e.  RR )
1002, 99ifclda 3976 . 2  |-  ( s  e.  ( -u pi [,] pi )  ->  if ( s  =  0 ,  1 ,  ( s  /  ( 2  x.  ( sin `  (
s  /  2 ) ) ) ) )  e.  RR )
1011, 100fmpti 6055 1  |-  K :
( -u pi [,] pi )
--> RR
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   ifcif 3944   class class class wbr 4456    |-> cmpt 4515   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    x. cmul 9514   RR*cxr 9644    < clt 9645    <_ cle 9646   -ucneg 9825    / cdiv 10227   2c2 10606   RR+crp 11245   (,)cioo 11554   [,)cico 11556   [,]cicc 11557   sincsin 13811   picpi 13814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-ef 13815  df-sin 13817  df-cos 13818  df-pi 13820  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-limc 22396  df-dv 22397
This theorem is referenced by:  fourierdlem55  32147  fourierdlem62  32154  fourierdlem66  32158  fourierdlem77  32169  fourierdlem85  32177  fourierdlem88  32180  fourierdlem103  32195  fourierdlem104  32196
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