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Theorem limccog 37276
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 22722 and limccnp 22720, 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 22704 . . 3  |-  ( G lim
CC  B )  C_  CC
2 limccog.3 . . 3  |-  ( ph  ->  C  e.  ( G lim
CC  B ) )
31, 2sseldi 3459 . 2  |-  ( ph  ->  C  e.  CC )
4 limcrcl 22703 . . . . . . . . . . . 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 1017 . . . . . . . . . 10  |-  ( ph  ->  G : dom  G --> CC )
75simp2d 1018 . . . . . . . . . 10  |-  ( ph  ->  dom  G  C_  CC )
85simp3d 1019 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
9 eqid 2420 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
106, 7, 8, 9ellimc2 22706 . . . . . . . . 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 213 . . . . . . . 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 464 . . . . . . 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 2792 . . . . . 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 430 . . . . 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 1068 . . . . . . . 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 1006 . . . . . . . 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 1033 . . . . . . . 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 22703 . . . . . . . . . . . . . . 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 1017 . . . . . . . . . . . . 13  |-  ( ph  ->  F : dom  F --> CC )
2220simp2d 1018 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  F  C_  CC )
2320simp3d 1019 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  CC )
2421, 22, 23, 9ellimc2 22706 . . . . . . . . . . . 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 213 . . . . . . . . . . 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 464 . . . . . . . . . 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 2792 . . . . . . . . 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 430 . . . . . . . 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 1263 . . . . . . 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 5351 . . . . . . . . . . 11  |-  ( ( G  o.  F )
" ( w  i^i  ( dom  F  \  { A } ) ) )  =  ( G
" ( F "
( w  i^i  ( dom  F  \  { A } ) ) ) )
3115ad2antrr 730 . . . . . . . . . . . 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 1061 . . . . . . . . . . . . 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 466 . . . . . . . . . . . 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 462 . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  v )  -> 
( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  v )
36 imassrn 5190 . . . . . . . . . . . . . . . . . 18  |-  ( F
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  ran  F
37 limccog.1 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ran  F  C_  ( dom  G  \  { B } ) )
3836, 37syl5ss 3472 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  ( dom  G  \  { B } ) )
3938adantr 466 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  v )  -> 
( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  ( dom  G  \  { B } ) )
4035, 39ssind 3683 . . . . . . . . . . . . . . 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 5215 . . . . . . . . . . . . . . 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 719 . . . . . . . . . . . . 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 760 . . . . . . . . . . . . 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 3471 . . . . . . . . . . . 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 1263 . . . . . . . . . . 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 3505 . . . . . . . . . 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 435 . . . . . . . . 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 567 . . . . . . . 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 2898 . . . . . . 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 2917 . . . . 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 435 . . 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 2837 . 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 5739 . . . . . . 7  |-  ( F : dom  F --> CC  ->  Fun 
F )
5721, 56syl 17 . . . . . 6  |-  ( ph  ->  Fun  F )
58 fdmrn 5752 . . . . . 6  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
5957, 58sylib 199 . . . . 5  |-  ( ph  ->  F : dom  F --> ran  F )
6037difss2d 3592 . . . . 5  |-  ( ph  ->  ran  F  C_  dom  G )
6159, 60fssd 5746 . . . 4  |-  ( ph  ->  F : dom  F --> dom  G )
62 fco 5747 . . . 4  |-  ( ( G : dom  G --> CC  /\  F : dom  F --> dom  G )  -> 
( G  o.  F
) : dom  F --> CC )
636, 61, 62syl2anc 665 . . 3  |-  ( ph  ->  ( G  o.  F
) : dom  F --> CC )
6463, 22, 23, 9ellimc2 22706 . 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 930 1  |-  ( ph  ->  C  e.  ( ( G  o.  F ) lim
CC  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    e. wcel 1867   A.wral 2773   E.wrex 2774    \ cdif 3430    i^i cin 3432    C_ wss 3433   {csn 3993   dom cdm 4845   ran crn 4846   "cima 4848    o. ccom 4849   Fun wfun 5586   -->wf 5588   ` cfv 5592  (class class class)co 6296   CCcc 9526   TopOpenctopn 15272  ℂfldccnfld 18898   lim CC climc 22691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fi 7922  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-fz 11772  df-seq 12200  df-exp 12259  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-plusg 15155  df-mulr 15156  df-starv 15157  df-tset 15161  df-ple 15162  df-ds 15164  df-unif 15165  df-rest 15273  df-topn 15274  df-topgen 15294  df-psmet 18890  df-xmet 18891  df-met 18892  df-bl 18893  df-mopn 18894  df-cnfld 18899  df-top 19845  df-bases 19846  df-topon 19847  df-topsp 19848  df-cnp 20168  df-xms 21259  df-ms 21260  df-limc 22695
This theorem is referenced by:  dirkercncflem2  37539  fourierdlem53  37595  fourierdlem93  37635  fourierdlem111  37653
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