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Theorem climcncf 21249
Description: Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)
Hypotheses
Ref Expression
climcncf.1  |-  Z  =  ( ZZ>= `  M )
climcncf.2  |-  ( ph  ->  M  e.  ZZ )
climcncf.4  |-  ( ph  ->  F  e.  ( A
-cn-> B ) )
climcncf.5  |-  ( ph  ->  G : Z --> A )
climcncf.6  |-  ( ph  ->  G  ~~>  D )
climcncf.7  |-  ( ph  ->  D  e.  A )
Assertion
Ref Expression
climcncf  |-  ( ph  ->  ( F  o.  G
)  ~~>  ( F `  D ) )

Proof of Theorem climcncf
Dummy variables  y 
z  x  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climcncf.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 climcncf.2 . 2  |-  ( ph  ->  M  e.  ZZ )
3 climcncf.7 . 2  |-  ( ph  ->  D  e.  A )
4 climcncf.4 . . . . 5  |-  ( ph  ->  F  e.  ( A
-cn-> B ) )
5 cncff 21242 . . . . 5  |-  ( F  e.  ( A -cn-> B )  ->  F : A
--> B )
64, 5syl 16 . . . 4  |-  ( ph  ->  F : A --> B )
76ffvelrnda 6031 . . 3  |-  ( (
ph  /\  z  e.  A )  ->  ( F `  z )  e.  B )
8 cncfrss2 21241 . . . . 5  |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
94, 8syl 16 . . . 4  |-  ( ph  ->  B  C_  CC )
109sselda 3509 . . 3  |-  ( (
ph  /\  ( F `  z )  e.  B
)  ->  ( F `  z )  e.  CC )
117, 10syldan 470 . 2  |-  ( (
ph  /\  z  e.  A )  ->  ( F `  z )  e.  CC )
12 climcncf.6 . 2  |-  ( ph  ->  G  ~~>  D )
13 climcncf.5 . . . 4  |-  ( ph  ->  G : Z --> A )
14 fvex 5881 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
151, 14eqeltri 2551 . . . 4  |-  Z  e. 
_V
16 fex 6143 . . . 4  |-  ( ( G : Z --> A  /\  Z  e.  _V )  ->  G  e.  _V )
1713, 15, 16sylancl 662 . . 3  |-  ( ph  ->  G  e.  _V )
18 coexg 6745 . . 3  |-  ( ( F  e.  ( A
-cn-> B )  /\  G  e.  _V )  ->  ( F  o.  G )  e.  _V )
194, 17, 18syl2anc 661 . 2  |-  ( ph  ->  ( F  o.  G
)  e.  _V )
20 cncfi 21243 . . . . 5  |-  ( ( F  e.  ( A
-cn-> B )  /\  D  e.  A  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  A  ( ( abs `  ( z  -  D
) )  <  y  ->  ( abs `  (
( F `  z
)  -  ( F `
 D ) ) )  <  x ) )
21203expia 1198 . . . 4  |-  ( ( F  e.  ( A
-cn-> B )  /\  D  e.  A )  ->  (
x  e.  RR+  ->  E. y  e.  RR+  A. z  e.  A  ( ( abs `  ( z  -  D ) )  < 
y  ->  ( abs `  ( ( F `  z )  -  ( F `  D )
) )  <  x
) ) )
224, 3, 21syl2anc 661 . . 3  |-  ( ph  ->  ( x  e.  RR+  ->  E. y  e.  RR+  A. z  e.  A  ( ( abs `  (
z  -  D ) )  <  y  -> 
( abs `  (
( F `  z
)  -  ( F `
 D ) ) )  <  x ) ) )
2322imp 429 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  A  ( ( abs `  ( z  -  D
) )  <  y  ->  ( abs `  (
( F `  z
)  -  ( F `
 D ) ) )  <  x ) )
2413ffvelrnda 6031 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  A )
25 fvco3 5950 . . 3  |-  ( ( G : Z --> A  /\  k  e.  Z )  ->  ( ( F  o.  G ) `  k
)  =  ( F `
 ( G `  k ) ) )
2613, 25sylan 471 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F  o.  G
) `  k )  =  ( F `  ( G `  k ) ) )
271, 2, 3, 11, 12, 19, 23, 24, 26climcn1 13389 1  |-  ( ph  ->  ( F  o.  G
)  ~~>  ( F `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   _Vcvv 3118    C_ wss 3481   class class class wbr 4452    o. ccom 5008   -->wf 5589   ` cfv 5593  (class class class)co 6294   CCcc 9500    < clt 9638    - cmin 9815   ZZcz 10874   ZZ>=cuz 11092   RR+crp 11230   abscabs 13042    ~~> cli 13282   -cn->ccncf 21225
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
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-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-po 4805  df-so 4806  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-2 10604  df-z 10875  df-uz 11093  df-cj 12907  df-re 12908  df-im 12909  df-abs 13044  df-clim 13286  df-cncf 21227
This theorem is referenced by:  leibpi  23116  lgamcvg2  28390  gamcvg  28391  iprodefisum  29019  climexp  31438  stirlinglem14  31678
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