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Theorem limccog 37694
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 22841 and limccnp 22839, 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 22823 . . 3  |-  ( G lim
CC  B )  C_  CC
2 limccog.3 . . 3  |-  ( ph  ->  C  e.  ( G lim
CC  B ) )
31, 2sseldi 3429 . 2  |-  ( ph  ->  C  e.  CC )
4 limcrcl 22822 . . . . . . . . . . . 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 1019 . . . . . . . . . 10  |-  ( ph  ->  G : dom  G --> CC )
75simp2d 1020 . . . . . . . . . 10  |-  ( ph  ->  dom  G  C_  CC )
85simp3d 1021 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
9 eqid 2450 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
106, 7, 8, 9ellimc2 22825 . . . . . . . . 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 214 . . . . . . . 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 465 . . . . . . 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 2756 . . . . . 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 431 . . . . 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 1070 . . . . . . . 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 1008 . . . . . . . 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 1035 . . . . . . . 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 22822 . . . . . . . . . . . . . . 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 1019 . . . . . . . . . . . . 13  |-  ( ph  ->  F : dom  F --> CC )
2220simp2d 1020 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  F  C_  CC )
2320simp3d 1021 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  CC )
2421, 22, 23, 9ellimc2 22825 . . . . . . . . . . . 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 214 . . . . . . . . . . 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 465 . . . . . . . . . 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 2756 . . . . . . . . 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 431 . . . . . . . 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 1266 . . . . . . 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 5339 . . . . . . . . . . 11  |-  ( ( G  o.  F )
" ( w  i^i  ( dom  F  \  { A } ) ) )  =  ( G
" ( F "
( w  i^i  ( dom  F  \  { A } ) ) ) )
3115ad2antrr 731 . . . . . . . . . . . 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 1063 . . . . . . . . . . . . 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 467 . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  v )  -> 
( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  v )
36 imassrn 5178 . . . . . . . . . . . . . . . . . 18  |-  ( F
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  ran  F
37 limccog.1 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ran  F  C_  ( dom  G  \  { B } ) )
3836, 37syl5ss 3442 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  ( dom  G  \  { B } ) )
3938adantr 467 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  v )  -> 
( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  ( dom  G  \  { B } ) )
4035, 39ssind 3655 . . . . . . . . . . . . . . 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 5203 . . . . . . . . . . . . . . 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 720 . . . . . . . . . . . . 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 761 . . . . . . . . . . . . 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 3441 . . . . . . . . . . . 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 1266 . . . . . . . . . . 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 3475 . . . . . . . . . 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 436 . . . . . . . . 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 568 . . . . . . . 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 2861 . . . . . . 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 2880 . . . . 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 436 . . 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 2801 . 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 5729 . . . . . . 7  |-  ( F : dom  F --> CC  ->  Fun 
F )
5721, 56syl 17 . . . . . 6  |-  ( ph  ->  Fun  F )
58 fdmrn 5742 . . . . . 6  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
5957, 58sylib 200 . . . . 5  |-  ( ph  ->  F : dom  F --> ran  F )
6037difss2d 3562 . . . . 5  |-  ( ph  ->  ran  F  C_  dom  G )
6159, 60fssd 5736 . . . 4  |-  ( ph  ->  F : dom  F --> dom  G )
62 fco 5737 . . . 4  |-  ( ( G : dom  G --> CC  /\  F : dom  F --> dom  G )  -> 
( G  o.  F
) : dom  F --> CC )
636, 61, 62syl2anc 666 . . 3  |-  ( ph  ->  ( G  o.  F
) : dom  F --> CC )
6463, 22, 23, 9ellimc2 22825 . 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 932 1  |-  ( ph  ->  C  e.  ( ( G  o.  F ) lim
CC  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 984    e. wcel 1886   A.wral 2736   E.wrex 2737    \ cdif 3400    i^i cin 3402    C_ wss 3403   {csn 3967   dom cdm 4833   ran crn 4834   "cima 4836    o. ccom 4837   Fun wfun 5575   -->wf 5577   ` cfv 5581  (class class class)co 6288   CCcc 9534   TopOpenctopn 15313  ℂfldccnfld 18963   lim CC climc 22810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fi 7922  df-sup 7953  df-inf 7954  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-fz 11782  df-seq 12211  df-exp 12270  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-plusg 15196  df-mulr 15197  df-starv 15198  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-rest 15314  df-topn 15315  df-topgen 15335  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cnp 20237  df-xms 21328  df-ms 21329  df-limc 22814
This theorem is referenced by:  dirkercncflem2  37960  fourierdlem53  38017  fourierdlem93  38057  fourierdlem111  38075
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