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Theorem climcncf 18930
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 18923 . . . . 5  |-  ( F  e.  ( A -cn-> B )  ->  F : A
--> B )
64, 5syl 16 . . . 4  |-  ( ph  ->  F : A --> B )
76ffvelrnda 5870 . . 3  |-  ( (
ph  /\  z  e.  A )  ->  ( F `  z )  e.  B )
8 cncfrss2 18922 . . . . 5  |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
94, 8syl 16 . . . 4  |-  ( ph  ->  B  C_  CC )
109sselda 3348 . . 3  |-  ( (
ph  /\  ( F `  z )  e.  B
)  ->  ( F `  z )  e.  CC )
117, 10syldan 457 . 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 5742 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
151, 14eqeltri 2506 . . . 4  |-  Z  e. 
_V
16 fex 5969 . . . 4  |-  ( ( G : Z --> A  /\  Z  e.  _V )  ->  G  e.  _V )
1713, 15, 16sylancl 644 . . 3  |-  ( ph  ->  G  e.  _V )
18 coexg 5412 . . 3  |-  ( ( F  e.  ( A
-cn-> B )  /\  G  e.  _V )  ->  ( F  o.  G )  e.  _V )
194, 17, 18syl2anc 643 . 2  |-  ( ph  ->  ( F  o.  G
)  e.  _V )
20 cncfi 18924 . . . . 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 1155 . . . 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 643 . . 3  |-  ( ph  ->  ( x  e.  RR+  ->  E. y  e.  RR+  A. z  e.  A  ( ( abs `  (
z  -  D ) )  <  y  -> 
( abs `  (
( F `  z
)  -  ( F `
 D ) ) )  <  x ) ) )
2322imp 419 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  A  ( ( abs `  ( z  -  D
) )  <  y  ->  ( abs `  (
( F `  z
)  -  ( F `
 D ) ) )  <  x ) )
2413ffvelrnda 5870 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  A )
25 fvco3 5800 . . 3  |-  ( ( G : Z --> A  /\  k  e.  Z )  ->  ( ( F  o.  G ) `  k
)  =  ( F `
 ( G `  k ) ) )
2613, 25sylan 458 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F  o.  G
) `  k )  =  ( F `  ( G `  k ) ) )
271, 2, 3, 11, 12, 19, 23, 24, 26climcn1 12385 1  |-  ( ph  ->  ( F  o.  G
)  ~~>  ( F `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   _Vcvv 2956    C_ wss 3320   class class class wbr 4212    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988    < clt 9120    - cmin 9291   ZZcz 10282   ZZ>=cuz 10488   RR+crp 10612   abscabs 12039    ~~> cli 12278   -cn->ccncf 18906
This theorem is referenced by:  leibpi  20782  lgamcvg2  24839  gamcvg  24840  iprodefisum  25318  climexp  27707  stirlinglem14  27812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-2 10058  df-z 10283  df-uz 10489  df-cj 11904  df-re 11905  df-im 11906  df-abs 12041  df-clim 12282  df-cncf 18908
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