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Theorem limccog 37797
Description: Limit of the composition of two functions. If the limit of 
F at  A is  B and the limit of  G at  B is  C, then the limit of  G  o.  F at  A is  C. With respect to limcco 22927 and limccnp 22925, here we drop continuity assumptions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
limccog.1  |-  ( ph  ->  ran  F  C_  ( dom  G  \  { B } ) )
limccog.2  |-  ( ph  ->  B  e.  ( F lim
CC  A ) )
limccog.3  |-  ( ph  ->  C  e.  ( G lim
CC  B ) )
Assertion
Ref Expression
limccog  |-  ( ph  ->  C  e.  ( ( G  o.  F ) lim
CC  A ) )

Proof of Theorem limccog
Dummy variables  u  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limccl 22909 . . 3  |-  ( G lim
CC  B )  C_  CC
2 limccog.3 . . 3  |-  ( ph  ->  C  e.  ( G lim
CC  B ) )
31, 2sseldi 3416 . 2  |-  ( ph  ->  C  e.  CC )
4 limcrcl 22908 . . . . . . . . . . . 12  |-  ( C  e.  ( G lim CC  B )  ->  ( G : dom  G --> CC  /\  dom  G  C_  CC  /\  B  e.  CC ) )
52, 4syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( G : dom  G --> CC  /\  dom  G  C_  CC  /\  B  e.  CC ) )
65simp1d 1042 . . . . . . . . . 10  |-  ( ph  ->  G : dom  G --> CC )
75simp2d 1043 . . . . . . . . . 10  |-  ( ph  ->  dom  G  C_  CC )
85simp3d 1044 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
9 eqid 2471 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
106, 7, 8, 9ellimc2 22911 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  ( G lim CC  B )  <-> 
( C  e.  CC  /\ 
A. u  e.  (
TopOpen ` fld ) ( C  e.  u  ->  E. v  e.  ( TopOpen ` fld ) ( B  e.  v  /\  ( G
" ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u )
) ) ) )
112, 10mpbid 215 . . . . . . . 8  |-  ( ph  ->  ( C  e.  CC  /\ 
A. u  e.  (
TopOpen ` fld ) ( C  e.  u  ->  E. v  e.  ( TopOpen ` fld ) ( B  e.  v  /\  ( G
" ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u )
) ) )
1211simprd 470 . . . . . . 7  |-  ( ph  ->  A. u  e.  (
TopOpen ` fld ) ( C  e.  u  ->  E. v  e.  ( TopOpen ` fld ) ( B  e.  v  /\  ( G
" ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u )
) )
1312r19.21bi 2776 . . . . . 6  |-  ( (
ph  /\  u  e.  ( TopOpen ` fld ) )  ->  ( C  e.  u  ->  E. v  e.  ( TopOpen ` fld )
( B  e.  v  /\  ( G "
( v  i^i  ( dom  G  \  { B } ) ) ) 
C_  u ) ) )
1413imp 436 . . . . 5  |-  ( ( ( ph  /\  u  e.  ( TopOpen ` fld ) )  /\  C  e.  u )  ->  E. v  e.  ( TopOpen ` fld ) ( B  e.  v  /\  ( G
" ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u )
)
15 simp1ll 1093 . . . . . . . 8  |-  ( ( ( ( ph  /\  u  e.  ( TopOpen ` fld )
)  /\  C  e.  u )  /\  v  e.  ( TopOpen ` fld )  /\  ( B  e.  v  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u
) )  ->  ph )
16 simp2 1031 . . . . . . . 8  |-  ( ( ( ( ph  /\  u  e.  ( TopOpen ` fld )
)  /\  C  e.  u )  /\  v  e.  ( TopOpen ` fld )  /\  ( B  e.  v  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u
) )  ->  v  e.  ( TopOpen ` fld ) )
17 simp3l 1058 . . . . . . . 8  |-  ( ( ( ( ph  /\  u  e.  ( TopOpen ` fld )
)  /\  C  e.  u )  /\  v  e.  ( TopOpen ` fld )  /\  ( B  e.  v  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u
) )  ->  B  e.  v )
18 limccog.2 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  ( F lim
CC  A ) )
19 limcrcl 22908 . . . . . . . . . . . . . . 15  |-  ( B  e.  ( F lim CC  A )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  A  e.  CC ) )
2018, 19syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  A  e.  CC ) )
2120simp1d 1042 . . . . . . . . . . . . 13  |-  ( ph  ->  F : dom  F --> CC )
2220simp2d 1043 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  F  C_  CC )
2320simp3d 1044 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  CC )
2421, 22, 23, 9ellimc2 22911 . . . . . . . . . . . 12  |-  ( ph  ->  ( B  e.  ( F lim CC  A )  <-> 
( B  e.  CC  /\ 
A. v  e.  (
TopOpen ` fld ) ( B  e.  v  ->  E. w  e.  ( TopOpen ` fld ) ( A  e.  w  /\  ( F
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  v )
) ) ) )
2518, 24mpbid 215 . . . . . . . . . . 11  |-  ( ph  ->  ( B  e.  CC  /\ 
A. v  e.  (
TopOpen ` fld ) ( B  e.  v  ->  E. w  e.  ( TopOpen ` fld ) ( A  e.  w  /\  ( F
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  v )
) ) )
2625simprd 470 . . . . . . . . . 10  |-  ( ph  ->  A. v  e.  (
TopOpen ` fld ) ( B  e.  v  ->  E. w  e.  ( TopOpen ` fld ) ( A  e.  w  /\  ( F
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  v )
) )
2726r19.21bi 2776 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( TopOpen ` fld ) )  ->  ( B  e.  v  ->  E. w  e.  ( TopOpen ` fld )
( A  e.  w  /\  ( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  v ) ) )
2827imp 436 . . . . . . . 8  |-  ( ( ( ph  /\  v  e.  ( TopOpen ` fld ) )  /\  B  e.  v )  ->  E. w  e.  ( TopOpen ` fld ) ( A  e.  w  /\  ( F
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  v )
)
2915, 16, 17, 28syl21anc 1291 . . . . . . 7  |-  ( ( ( ( ph  /\  u  e.  ( TopOpen ` fld )
)  /\  C  e.  u )  /\  v  e.  ( TopOpen ` fld )  /\  ( B  e.  v  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u
) )  ->  E. w  e.  ( TopOpen ` fld ) ( A  e.  w  /\  ( F
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  v )
)
30 imaco 5347 . . . . . . . . . . 11  |-  ( ( G  o.  F )
" ( w  i^i  ( dom  F  \  { A } ) ) )  =  ( G
" ( F "
( w  i^i  ( dom  F  \  { A } ) ) ) )
3115ad2antrr 740 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  u  e.  ( TopOpen ` fld ) )  /\  C  e.  u )  /\  v  e.  ( TopOpen ` fld )  /\  ( B  e.  v  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u
) )  /\  w  e.  ( TopOpen ` fld ) )  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) )  C_  v )  ->  ph )
32 simpl3r 1086 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  u  e.  ( TopOpen ` fld )
)  /\  C  e.  u )  /\  v  e.  ( TopOpen ` fld )  /\  ( B  e.  v  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u
) )  /\  w  e.  ( TopOpen ` fld ) )  ->  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u )
3332adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  u  e.  ( TopOpen ` fld ) )  /\  C  e.  u )  /\  v  e.  ( TopOpen ` fld )  /\  ( B  e.  v  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u
) )  /\  w  e.  ( TopOpen ` fld ) )  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) )  C_  v )  ->  ( G " (
v  i^i  ( dom  G 
\  { B }
) ) )  C_  u )
34 simpr 468 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  u  e.  ( TopOpen ` fld ) )  /\  C  e.  u )  /\  v  e.  ( TopOpen ` fld )  /\  ( B  e.  v  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u
) )  /\  w  e.  ( TopOpen ` fld ) )  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) )  C_  v )  ->  ( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  v )
35 simpr 468 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  v )  -> 
( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  v )
36 imassrn 5185 . . . . . . . . . . . . . . . . . 18  |-  ( F
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  ran  F
37 limccog.1 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ran  F  C_  ( dom  G  \  { B } ) )
3836, 37syl5ss 3429 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  ( dom  G  \  { B } ) )
3938adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  v )  -> 
( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  ( dom  G  \  { B } ) )
4035, 39ssind 3647 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  v )  -> 
( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  ( v  i^i  ( dom  G  \  { B } ) ) )
41 imass2 5210 . . . . . . . . . . . . . . 15  |-  ( ( F " ( w  i^i  ( dom  F  \  { A } ) ) )  C_  (
v  i^i  ( dom  G 
\  { B }
) )  ->  ( G " ( F "
( w  i^i  ( dom  F  \  { A } ) ) ) )  C_  ( G " ( v  i^i  ( dom  G  \  { B } ) ) ) )
4240, 41syl 17 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  v )  -> 
( G " ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) )  C_  ( G " ( v  i^i  ( dom  G  \  { B } ) ) ) )
4342adantlr 729 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u )  /\  ( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  v )  ->  ( G " ( F "
( w  i^i  ( dom  F  \  { A } ) ) ) )  C_  ( G " ( v  i^i  ( dom  G  \  { B } ) ) ) )
44 simplr 770 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u )  /\  ( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  v )  ->  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u )
4543, 44sstrd 3428 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u )  /\  ( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  v )  ->  ( G " ( F "
( w  i^i  ( dom  F  \  { A } ) ) ) )  C_  u )
4631, 33, 34, 45syl21anc 1291 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  u  e.  ( TopOpen ` fld ) )  /\  C  e.  u )  /\  v  e.  ( TopOpen ` fld )  /\  ( B  e.  v  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u
) )  /\  w  e.  ( TopOpen ` fld ) )  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) )  C_  v )  ->  ( G " ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) )  C_  u
)
4730, 46syl5eqss 3462 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  u  e.  ( TopOpen ` fld ) )  /\  C  e.  u )  /\  v  e.  ( TopOpen ` fld )  /\  ( B  e.  v  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u
) )  /\  w  e.  ( TopOpen ` fld ) )  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) )  C_  v )  ->  ( ( G  o.  F ) " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  u )
4847ex 441 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  u  e.  ( TopOpen ` fld )
)  /\  C  e.  u )  /\  v  e.  ( TopOpen ` fld )  /\  ( B  e.  v  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u
) )  /\  w  e.  ( TopOpen ` fld ) )  ->  (
( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  v  ->  ( ( G  o.  F ) "
( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  u ) )
4948anim2d 575 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  u  e.  ( TopOpen ` fld )
)  /\  C  e.  u )  /\  v  e.  ( TopOpen ` fld )  /\  ( B  e.  v  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u
) )  /\  w  e.  ( TopOpen ` fld ) )  ->  (
( A  e.  w  /\  ( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  v )  ->  ( A  e.  w  /\  ( ( G  o.  F ) " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  u ) ) )
5049reximdva 2858 . . . . . . 7  |-  ( ( ( ( ph  /\  u  e.  ( TopOpen ` fld )
)  /\  C  e.  u )  /\  v  e.  ( TopOpen ` fld )  /\  ( B  e.  v  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u
) )  ->  ( E. w  e.  ( TopOpen
` fld
) ( A  e.  w  /\  ( F
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  v )  ->  E. w  e.  (
TopOpen ` fld ) ( A  e.  w  /\  ( ( G  o.  F )
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  u )
) )
5129, 50mpd 15 . . . . . 6  |-  ( ( ( ( ph  /\  u  e.  ( TopOpen ` fld )
)  /\  C  e.  u )  /\  v  e.  ( TopOpen ` fld )  /\  ( B  e.  v  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u
) )  ->  E. w  e.  ( TopOpen ` fld ) ( A  e.  w  /\  ( ( G  o.  F )
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  u )
)
5251rexlimdv3a 2873 . . . . 5  |-  ( ( ( ph  /\  u  e.  ( TopOpen ` fld ) )  /\  C  e.  u )  ->  ( E. v  e.  ( TopOpen
` fld
) ( B  e.  v  /\  ( G
" ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u )  ->  E. w  e.  (
TopOpen ` fld ) ( A  e.  w  /\  ( ( G  o.  F )
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  u )
) )
5314, 52mpd 15 . . . 4  |-  ( ( ( ph  /\  u  e.  ( TopOpen ` fld ) )  /\  C  e.  u )  ->  E. w  e.  ( TopOpen ` fld ) ( A  e.  w  /\  ( ( G  o.  F )
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  u )
)
5453ex 441 . . 3  |-  ( (
ph  /\  u  e.  ( TopOpen ` fld ) )  ->  ( C  e.  u  ->  E. w  e.  ( TopOpen ` fld )
( A  e.  w  /\  ( ( G  o.  F ) " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  u ) ) )
5554ralrimiva 2809 . 2  |-  ( ph  ->  A. u  e.  (
TopOpen ` fld ) ( C  e.  u  ->  E. w  e.  ( TopOpen ` fld ) ( A  e.  w  /\  ( ( G  o.  F )
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  u )
) )
56 ffun 5742 . . . . . . 7  |-  ( F : dom  F --> CC  ->  Fun 
F )
5721, 56syl 17 . . . . . 6  |-  ( ph  ->  Fun  F )
58 fdmrn 5756 . . . . . 6  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
5957, 58sylib 201 . . . . 5  |-  ( ph  ->  F : dom  F --> ran  F )
6037difss2d 3552 . . . . 5  |-  ( ph  ->  ran  F  C_  dom  G )
6159, 60fssd 5750 . . . 4  |-  ( ph  ->  F : dom  F --> dom  G )
62 fco 5751 . . . 4  |-  ( ( G : dom  G --> CC  /\  F : dom  F --> dom  G )  -> 
( G  o.  F
) : dom  F --> CC )
636, 61, 62syl2anc 673 . . 3  |-  ( ph  ->  ( G  o.  F
) : dom  F --> CC )
6463, 22, 23, 9ellimc2 22911 . 2  |-  ( ph  ->  ( C  e.  ( ( G  o.  F
) lim CC  A )  <->  ( C  e.  CC  /\  A. u  e.  ( TopOpen ` fld )
( C  e.  u  ->  E. w  e.  (
TopOpen ` fld ) ( A  e.  w  /\  ( ( G  o.  F )
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  u )
) ) ) )
653, 55, 64mpbir2and 936 1  |-  ( ph  ->  C  e.  ( ( G  o.  F ) lim
CC  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    e. wcel 1904   A.wral 2756   E.wrex 2757    \ cdif 3387    i^i cin 3389    C_ wss 3390   {csn 3959   dom cdm 4839   ran crn 4840   "cima 4842    o. ccom 4843   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   TopOpenctopn 15398  ℂfldccnfld 19047   lim CC climc 22896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-fz 11811  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cnp 20321  df-xms 21413  df-ms 21414  df-limc 22900
This theorem is referenced by:  dirkercncflem2  38078  fourierdlem53  38135  fourierdlem93  38175  fourierdlem111  38193
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