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Theorem climcn1lem 10703
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 10666 . . 3  |-  ( F  ~~>  A  ->  A  e.  CC )
53, 4syl 14 . 2  |-  ( ph  ->  A  e.  CC )
6 climcn1lem.7 . . . 4  |-  H : CC
--> CC
76ffvelrni 5433 . . 3  |-  ( z  e.  CC  ->  ( H `  z )  e.  CC )
87adantl 271 . 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 277 . 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 10693 1  |-  ( ph  ->  G  ~~>  ( H `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   class class class wbr 3845   -->wf 5011   ` cfv 5015  (class class class)co 5652   CCcc 7346    < clt 7520    - cmin 7651   ZZcz 8748   ZZ>=cuz 9017   RR+crp 9132   abscabs 10426    ~~> cli 10662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-inn 8421  df-n0 8672  df-z 8749  df-uz 9018  df-clim 10663
This theorem is referenced by:  climabs  10704  climcj  10705  climre  10706  climim  10707
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