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Theorem climcncf 18420
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 18413 . . . . 5  |-  ( F  e.  ( A -cn-> B )  ->  F : A
--> B )
64, 5syl 15 . . . 4  |-  ( ph  ->  F : A --> B )
7 ffvelrn 5679 . . . 4  |-  ( ( F : A --> B  /\  z  e.  A )  ->  ( F `  z
)  e.  B )
86, 7sylan 457 . . 3  |-  ( (
ph  /\  z  e.  A )  ->  ( F `  z )  e.  B )
9 cncfrss2 18412 . . . . 5  |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
104, 9syl 15 . . . 4  |-  ( ph  ->  B  C_  CC )
1110sselda 3193 . . 3  |-  ( (
ph  /\  ( F `  z )  e.  B
)  ->  ( F `  z )  e.  CC )
128, 11syldan 456 . 2  |-  ( (
ph  /\  z  e.  A )  ->  ( F `  z )  e.  CC )
13 climcncf.6 . 2  |-  ( ph  ->  G  ~~>  D )
14 climcncf.5 . . . 4  |-  ( ph  ->  G : Z --> A )
15 fvex 5555 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
161, 15eqeltri 2366 . . . 4  |-  Z  e. 
_V
17 fex 5765 . . . 4  |-  ( ( G : Z --> A  /\  Z  e.  _V )  ->  G  e.  _V )
1814, 16, 17sylancl 643 . . 3  |-  ( ph  ->  G  e.  _V )
19 coexg 5231 . . 3  |-  ( ( F  e.  ( A
-cn-> B )  /\  G  e.  _V )  ->  ( F  o.  G )  e.  _V )
204, 18, 19syl2anc 642 . 2  |-  ( ph  ->  ( F  o.  G
)  e.  _V )
21 cncfi 18414 . . . . 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 ) )
22213expia 1153 . . . 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
) ) )
234, 3, 22syl2anc 642 . . 3  |-  ( ph  ->  ( x  e.  RR+  ->  E. y  e.  RR+  A. z  e.  A  ( ( abs `  (
z  -  D ) )  <  y  -> 
( abs `  (
( F `  z
)  -  ( F `
 D ) ) )  <  x ) ) )
2423imp 418 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  A  ( ( abs `  ( z  -  D
) )  <  y  ->  ( abs `  (
( F `  z
)  -  ( F `
 D ) ) )  <  x ) )
25 ffvelrn 5679 . . 3  |-  ( ( G : Z --> A  /\  k  e.  Z )  ->  ( G `  k
)  e.  A )
2614, 25sylan 457 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  A )
27 fvco3 5612 . . 3  |-  ( ( G : Z --> A  /\  k  e.  Z )  ->  ( ( F  o.  G ) `  k
)  =  ( F `
 ( G `  k ) ) )
2814, 27sylan 457 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F  o.  G
) `  k )  =  ( F `  ( G `  k ) ) )
291, 2, 3, 12, 13, 20, 24, 26, 28climcn1 12081 1  |-  ( ph  ->  ( F  o.  G
)  ~~>  ( F `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751    < clt 8883    - cmin 9053   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   abscabs 11735    ~~> cli 11974   -cn->ccncf 18396
This theorem is referenced by:  leibpi  20254  climexp  27834  stirlinglem14  27939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-z 10041  df-uz 10247  df-cj 11600  df-re 11601  df-im 11602  df-abs 11737  df-clim 11978  df-cncf 18398
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