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Theorem climcn1lem 10545
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 10509 . . 3  |-  ( F  ~~>  A  ->  A  e.  CC )
53, 4syl 14 . 2  |-  ( ph  ->  A  e.  CC )
6 climcn1lem.7 . . . 4  |-  H : CC
--> CC
76ffvelrni 5381 . . 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 10535 1  |-  ( ph  ->  G  ~~>  ( H `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1287    e. wcel 1436   A.wral 2355   E.wrex 2356   class class class wbr 3814   -->wf 4968   ` cfv 4972  (class class class)co 5594   CCcc 7269    < clt 7443    - cmin 7574   ZZcz 8660   ZZ>=cuz 8928   RR+crp 9043   abscabs 10271    ~~> cli 10505
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-addcom 7366  ax-addass 7368  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-0id 7374  ax-rnegex 7375  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-apti 7381  ax-pre-ltadd 7382
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-if 3377  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-inn 8335  df-n0 8584  df-z 8661  df-uz 8929  df-clim 10506
This theorem is referenced by:  climabs  10546  climcj  10547  climre  10548  climim  10549
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