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Theorem climcn1lem 11282
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 11245 . . 3  |-  ( F  ~~>  A  ->  A  e.  CC )
53, 4syl 14 . 2  |-  ( ph  ->  A  e.  CC )
6 climcn1lem.7 . . . 4  |-  H : CC
--> CC
76ffvelrni 5630 . . 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 11271 1  |-  ( ph  ->  G  ~~>  ( H `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   A.wral 2448   E.wrex 2449   class class class wbr 3989   -->wf 5194   ` cfv 5198  (class class class)co 5853   CCcc 7772    < clt 7954    - cmin 8090   ZZcz 9212   ZZ>=cuz 9487   RR+crp 9610   abscabs 10961    ~~> cli 11241
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-clim 11242
This theorem is referenced by:  climabs  11283  climcj  11284  climre  11285  climim  11286
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