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Theorem climcn1lem 11095
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 11058 . . 3  |-  ( F  ~~>  A  ->  A  e.  CC )
53, 4syl 14 . 2  |-  ( ph  ->  A  e.  CC )
6 climcn1lem.7 . . . 4  |-  H : CC
--> CC
76ffvelrni 5554 . . 3  |-  ( z  e.  CC  ->  ( H `  z )  e.  CC )
87adantl 275 . 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 281 . 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 11084 1  |-  ( ph  ->  G  ~~>  ( H `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   CCcc 7625    < clt 7807    - cmin 7940   ZZcz 9061   ZZ>=cuz 9333   RR+crp 9448   abscabs 10776    ~~> cli 11054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334  df-clim 11055
This theorem is referenced by:  climabs  11096  climcj  11097  climre  11098  climim  11099
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