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Theorem climcn1lem 12388
Description: The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.)
Hypotheses
Ref Expression
climcn1lem.1  |-  Z  =  ( ZZ>= `  M )
climcn1lem.2  |-  ( ph  ->  F  ~~>  A )
climcn1lem.4  |-  ( ph  ->  G  e.  W )
climcn1lem.5  |-  ( ph  ->  M  e.  ZZ )
climcn1lem.6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
climcn1lem.7  |-  H : CC
--> CC
climcn1lem.8  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  ( ( abs `  ( z  -  A
) )  <  y  ->  ( abs `  (
( H `  z
)  -  ( H `
 A ) ) )  <  x ) )
climcn1lem.9  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( H `  ( F `  k ) ) )
Assertion
Ref Expression
climcn1lem  |-  ( ph  ->  G  ~~>  ( H `  A ) )
Distinct variable groups:    x, k,
y, z, A    k, F, y, z    k, G, x    ph, k, x, y, z    k, Z, y   
k, H, x, y, z    k, M
Allowed substitution hints:    F( x)    G( y, z)    M( x, y, z)    W( x, y, z, k)    Z( x, z)

Proof of Theorem climcn1lem
StepHypRef Expression
1 climcn1lem.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 climcn1lem.5 . 2  |-  ( ph  ->  M  e.  ZZ )
3 climcn1lem.2 . . 3  |-  ( ph  ->  F  ~~>  A )
4 climcl 12285 . . 3  |-  ( F  ~~>  A  ->  A  e.  CC )
53, 4syl 16 . 2  |-  ( ph  ->  A  e.  CC )
6 climcn1lem.7 . . . 4  |-  H : CC
--> CC
76ffvelrni 5861 . . 3  |-  ( z  e.  CC  ->  ( H `  z )  e.  CC )
87adantl 453 . 2  |-  ( (
ph  /\  z  e.  CC )  ->  ( H `
 z )  e.  CC )
9 climcn1lem.4 . 2  |-  ( ph  ->  G  e.  W )
10 climcn1lem.8 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  ( ( abs `  ( z  -  A
) )  <  y  ->  ( abs `  (
( H `  z
)  -  ( H `
 A ) ) )  <  x ) )
115, 10sylan 458 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  ( ( abs `  ( z  -  A
) )  <  y  ->  ( abs `  (
( H `  z
)  -  ( H `
 A ) ) )  <  x ) )
12 climcn1lem.6 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
13 climcn1lem.9 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( H `  ( F `  k ) ) )
141, 2, 5, 8, 3, 9, 11, 12, 13climcn1 12377 1  |-  ( ph  ->  G  ~~>  ( H `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   class class class wbr 4204   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980    < clt 9112    - cmin 9283   ZZcz 10274   ZZ>=cuz 10480   RR+crp 10604   abscabs 12031    ~~> cli 12270
This theorem is referenced by:  climabs  12389  climcj  12390  climre  12391  climim  12392  sinccvglem  25101
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-neg 9286  df-z 10275  df-uz 10481  df-clim 12274
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