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Theorem climcncf 20618
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 20611 . . . . 5  |-  ( F  e.  ( A -cn-> B )  ->  F : A
--> B )
64, 5syl 16 . . . 4  |-  ( ph  ->  F : A --> B )
76ffvelrnda 5955 . . 3  |-  ( (
ph  /\  z  e.  A )  ->  ( F `  z )  e.  B )
8 cncfrss2 20610 . . . . 5  |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
94, 8syl 16 . . . 4  |-  ( ph  ->  B  C_  CC )
109sselda 3467 . . 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 5812 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
151, 14eqeltri 2538 . . . 4  |-  Z  e. 
_V
16 fex 6062 . . . 4  |-  ( ( G : Z --> A  /\  Z  e.  _V )  ->  G  e.  _V )
1713, 15, 16sylancl 662 . . 3  |-  ( ph  ->  G  e.  _V )
18 coexg 6641 . . 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 20612 . . . . 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 1190 . . . 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 5955 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  A )
25 fvco3 5880 . . 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 13191 1  |-  ( ph  ->  ( F  o.  G
)  ~~>  ( F `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800   _Vcvv 3078    C_ wss 3439   class class class wbr 4403    o. ccom 4955   -->wf 5525   ` cfv 5529  (class class class)co 6203   CCcc 9395    < clt 9533    - cmin 9710   ZZcz 10761   ZZ>=cuz 10976   RR+crp 11106   abscabs 12845    ~~> cli 13084   -cn->ccncf 20594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-2 10495  df-z 10762  df-uz 10977  df-cj 12710  df-re 12711  df-im 12712  df-abs 12847  df-clim 13088  df-cncf 20596
This theorem is referenced by:  leibpi  22480  lgamcvg2  27208  gamcvg  27209  iprodefisum  27672  climexp  29949  stirlinglem14  30053
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