Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  limccog Structured version   Unicode version

Theorem limccog 31485
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 22165 and limccnp 22163, 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 22147 . . . 4  |-  ( G lim
CC  B )  C_  CC
2 limccog.3 . . . 4  |-  ( ph  ->  C  e.  ( G lim
CC  B ) )
31, 2sseldi 3507 . . 3  |-  ( ph  ->  C  e.  CC )
4 limcrcl 22146 . . . . . . . . . . . . 13  |-  ( C  e.  ( G lim CC  B )  ->  ( G : dom  G --> CC  /\  dom  G  C_  CC  /\  B  e.  CC ) )
52, 4syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( G : dom  G --> CC  /\  dom  G  C_  CC  /\  B  e.  CC ) )
65simp1d 1008 . . . . . . . . . . 11  |-  ( ph  ->  G : dom  G --> CC )
75simp2d 1009 . . . . . . . . . . 11  |-  ( ph  ->  dom  G  C_  CC )
85simp3d 1010 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  CC )
9 eqid 2467 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
106, 7, 8, 9ellimc2 22149 . . . . . . . . . 10  |-  ( 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 210 . . . . . . . . 9  |-  ( 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 463 . . . . . . . 8  |-  ( 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 2836 . . . . . . 7  |-  ( (
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 429 . . . . . 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 )
)
15 simp1ll 1059 . . . . . . . . . . 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
) )  ->  ph )
16 simp2 997 . . . . . . . . . . 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
) )  ->  v  e.  ( TopOpen ` fld ) )
1715, 16jca 532 . . . . . . . . . 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
) )  ->  ( ph  /\  v  e.  (
TopOpen ` fld ) ) )
18 simp3l 1024 . . . . . . . . . 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
) )  ->  B  e.  v )
19 limccog.2 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  ( F lim
CC  A ) )
20 limcrcl 22146 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  ( F lim CC  A )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  A  e.  CC ) )
2119, 20syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  A  e.  CC ) )
2221simp1d 1008 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : dom  F --> CC )
2321simp2d 1009 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  F  C_  CC )
2421simp3d 1010 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  CC )
2522, 23, 24, 9ellimc2 22149 . . . . . . . . . . . . . 14  |-  ( 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 )
) ) ) )
2619, 25mpbid 210 . . . . . . . . . . . . 13  |-  ( 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 )
) ) )
2726simprd 463 . . . . . . . . . . . 12  |-  ( 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 )
) )
2827r19.21bi 2836 . . . . . . . . . . 11  |-  ( (
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 ) ) )
2928imp 429 . . . . . . . . . 10  |-  ( ( ( 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 )
)
3017, 18, 29syl2anc 661 . . . . . . . . 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
) )  ->  E. w  e.  ( TopOpen ` fld ) ( A  e.  w  /\  ( F
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  v )
)
3115ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
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 1052 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( 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 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
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 )
3431, 33jca 532 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
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  /\  ( G " ( v  i^i  ( dom  G  \  { B } ) ) )  C_  u )
)
35 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
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 )
36 simpr 461 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  v )  -> 
( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  v )
37 imassrn 5354 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  ran  F
38 limccog.1 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ran  F  C_  ( dom  G  \  { B } ) )
3937, 38syl5ss 3520 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  ( dom  G  \  { B } ) )
4039adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  v )  -> 
( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  ( dom  G  \  { B } ) )
4136, 40jca 532 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  v )  -> 
( ( F "
( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  v  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) )  C_  ( dom  G 
\  { B }
) ) )
42 ssin 3725 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  v  /\  ( F "
( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  ( dom  G  \  { B } ) )  <->  ( F "
( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  ( v  i^i  ( dom  G  \  { B } ) ) )
4341, 42sylib 196 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  v )  -> 
( F " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  ( v  i^i  ( dom  G  \  { B } ) ) )
44 imass2 5378 . . . . . . . . . . . . . . . . 17  |-  ( ( 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 } ) ) ) )
4543, 44syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
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 } ) ) ) )
4645adantlr 714 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 } ) ) ) )
47 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 )
4846, 47sstrd 3519 . . . . . . . . . . . . . 14  |-  ( ( ( 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 )
4934, 35, 48syl2anc 661 . . . . . . . . . . . . 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 ) )  /\  ( F " ( w  i^i  ( dom  F  \  { A } ) ) )  C_  v )  ->  ( G " ( F " ( w  i^i  ( dom  F  \  { A } ) ) ) )  C_  u
)
50 imaco 5518 . . . . . . . . . . . . . 14  |-  ( ( G  o.  F )
" ( w  i^i  ( dom  F  \  { A } ) ) )  =  ( G
" ( F "
( w  i^i  ( dom  F  \  { A } ) ) ) )
5150sseq1i 3533 . . . . . . . . . . . . 13  |-  ( ( ( G  o.  F
) " ( w  i^i  ( dom  F  \  { A } ) ) )  C_  u  <->  ( G " ( F
" ( w  i^i  ( dom  F  \  { A } ) ) ) )  C_  u
)
5249, 51sylibr 212 . . . . . . . . . . . 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  o.  F ) " (
w  i^i  ( dom  F 
\  { A }
) ) )  C_  u )
5352ex 434 . . . . . . . . . . 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  o.  F ) "
( w  i^i  ( dom  F  \  { A } ) ) ) 
C_  u ) )
5453anim2d 565 . . . . . . . . . 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 ) )  ->  (
( 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 ) ) )
5554reximdva 2942 . . . . . . . . 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
) )  ->  ( 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 )
) )
5630, 55mpd 15 . . . . . . . 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
) )  ->  E. w  e.  ( TopOpen ` fld ) ( A  e.  w  /\  ( ( G  o.  F )
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  u )
)
57563exp 1195 . . . . . . 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  /\  ( ( G  o.  F )
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  u )
) ) )
5857rexlimdv 2957 . . . . . 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 )  ->  E. w  e.  (
TopOpen ` fld ) ( A  e.  w  /\  ( ( G  o.  F )
" ( w  i^i  ( dom  F  \  { A } ) ) )  C_  u )
) )
5914, 58mpd 15 . . . . 5  |-  ( ( ( 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 )
)
6059ex 434 . . . 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 ) ) )
6160ralrimiva 2881 . . 3  |-  ( 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 )
) )
623, 61jca 532 . 2  |-  ( ph  ->  ( 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 )
) ) )
63 ffun 5739 . . . . . . 7  |-  ( F : dom  F --> CC  ->  Fun 
F )
6422, 63syl 16 . . . . . 6  |-  ( ph  ->  Fun  F )
65 fdmrn 5752 . . . . . 6  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
6664, 65sylib 196 . . . . 5  |-  ( ph  ->  F : dom  F --> ran  F )
6738difss2d 3639 . . . . 5  |-  ( ph  ->  ran  F  C_  dom  G )
68 fss 5745 . . . . 5  |-  ( ( F : dom  F --> ran  F  /\  ran  F  C_ 
dom  G )  ->  F : dom  F --> dom  G
)
6966, 67, 68syl2anc 661 . . . 4  |-  ( ph  ->  F : dom  F --> dom  G )
70 fco 5747 . . . 4  |-  ( ( G : dom  G --> CC  /\  F : dom  F --> dom  G )  -> 
( G  o.  F
) : dom  F --> CC )
716, 69, 70syl2anc 661 . . 3  |-  ( ph  ->  ( G  o.  F
) : dom  F --> CC )
7271, 23, 24, 9ellimc2 22149 . 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 )
) ) ) )
7362, 72mpbird 232 1  |-  ( ph  ->  C  e.  ( ( G  o.  F ) lim
CC  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767   A.wral 2817   E.wrex 2818    \ cdif 3478    i^i cin 3480    C_ wss 3481   {csn 4033   dom cdm 5005   ran crn 5006   "cima 5008    o. ccom 5009   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   TopOpenctopn 14694  ℂfldccnfld 18290   lim CC climc 22134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-fz 11685  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-plusg 14585  df-mulr 14586  df-starv 14587  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-rest 14695  df-topn 14696  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cnp 19597  df-xms 20691  df-ms 20692  df-limc 22138
This theorem is referenced by:  dirkercncflem2  31727  fourierdlem53  31783  fourierdlem93  31823  fourierdlem111  31841
  Copyright terms: Public domain W3C validator