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Theorem climcncf 21381
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 21374 . . . . 5  |-  ( F  e.  ( A -cn-> B )  ->  F : A
--> B )
64, 5syl 16 . . . 4  |-  ( ph  ->  F : A --> B )
76ffvelrnda 6016 . . 3  |-  ( (
ph  /\  z  e.  A )  ->  ( F `  z )  e.  B )
8 cncfrss2 21373 . . . . 5  |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
94, 8syl 16 . . . 4  |-  ( ph  ->  B  C_  CC )
109sselda 3489 . . 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 5866 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
151, 14eqeltri 2527 . . . 4  |-  Z  e. 
_V
16 fex 6130 . . . 4  |-  ( ( G : Z --> A  /\  Z  e.  _V )  ->  G  e.  _V )
1713, 15, 16sylancl 662 . . 3  |-  ( ph  ->  G  e.  _V )
18 coexg 6736 . . 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 21375 . . . . 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 1199 . . . 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 6016 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  A )
25 fvco3 5935 . . 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 13395 1  |-  ( ph  ->  ( F  o.  G
)  ~~>  ( F `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   _Vcvv 3095    C_ wss 3461   class class class wbr 4437    o. ccom 4993   -->wf 5574   ` cfv 5578  (class class class)co 6281   CCcc 9493    < clt 9631    - cmin 9810   ZZcz 10871   ZZ>=cuz 11091   RR+crp 11230   abscabs 13048    ~~> cli 13288   -cn->ccncf 21357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-2 10601  df-z 10872  df-uz 11092  df-cj 12913  df-re 12914  df-im 12915  df-abs 13050  df-clim 13292  df-cncf 21359
This theorem is referenced by:  leibpi  23249  lgamcvg2  28574  gamcvg  28575  iprodefisum  29099  climexp  31519  stirlinglem14  31758
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