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Theorem rlimi 12983
Description: Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 28-Feb-2015.)
Hypotheses
Ref Expression
rlimi.1  |-  ( ph  ->  A. z  e.  A  B  e.  V )
rlimi.2  |-  ( ph  ->  R  e.  RR+ )
rlimi.3  |-  ( ph  ->  ( z  e.  A  |->  B )  ~~> r  C
)
Assertion
Ref Expression
rlimi  |-  ( ph  ->  E. y  e.  RR  A. z  e.  A  ( y  <_  z  ->  ( abs `  ( B  -  C ) )  <  R ) )
Distinct variable groups:    y, z, A    y, B    y, C, z    ph, y    y, R, z    z, V
Allowed substitution hints:    ph( z)    B( z)    V( y)

Proof of Theorem rlimi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rlimi.2 . 2  |-  ( ph  ->  R  e.  RR+ )
2 rlimi.3 . . 3  |-  ( ph  ->  ( z  e.  A  |->  B )  ~~> r  C
)
3 rlimf 12971 . . . . . . 7  |-  ( ( z  e.  A  |->  B )  ~~> r  C  -> 
( z  e.  A  |->  B ) : dom  ( z  e.  A  |->  B ) --> CC )
42, 3syl 16 . . . . . 6  |-  ( ph  ->  ( z  e.  A  |->  B ) : dom  ( z  e.  A  |->  B ) --> CC )
5 rlimi.1 . . . . . . . . 9  |-  ( ph  ->  A. z  e.  A  B  e.  V )
6 eqid 2438 . . . . . . . . . 10  |-  ( z  e.  A  |->  B )  =  ( z  e.  A  |->  B )
76fmpt 5859 . . . . . . . . 9  |-  ( A. z  e.  A  B  e.  V  <->  ( z  e.  A  |->  B ) : A --> V )
85, 7sylib 196 . . . . . . . 8  |-  ( ph  ->  ( z  e.  A  |->  B ) : A --> V )
9 fdm 5558 . . . . . . . 8  |-  ( ( z  e.  A  |->  B ) : A --> V  ->  dom  ( z  e.  A  |->  B )  =  A )
108, 9syl 16 . . . . . . 7  |-  ( ph  ->  dom  ( z  e.  A  |->  B )  =  A )
1110feq2d 5542 . . . . . 6  |-  ( ph  ->  ( ( z  e.  A  |->  B ) : dom  ( z  e.  A  |->  B ) --> CC  <->  ( z  e.  A  |->  B ) : A --> CC ) )
124, 11mpbid 210 . . . . 5  |-  ( ph  ->  ( z  e.  A  |->  B ) : A --> CC )
136fmpt 5859 . . . . 5  |-  ( A. z  e.  A  B  e.  CC  <->  ( z  e.  A  |->  B ) : A --> CC )
1412, 13sylibr 212 . . . 4  |-  ( ph  ->  A. z  e.  A  B  e.  CC )
15 rlimss 12972 . . . . . 6  |-  ( ( z  e.  A  |->  B )  ~~> r  C  ->  dom  ( z  e.  A  |->  B )  C_  RR )
162, 15syl 16 . . . . 5  |-  ( ph  ->  dom  ( z  e.  A  |->  B )  C_  RR )
1710, 16eqsstr3d 3386 . . . 4  |-  ( ph  ->  A  C_  RR )
18 rlimcl 12973 . . . . 5  |-  ( ( z  e.  A  |->  B )  ~~> r  C  ->  C  e.  CC )
192, 18syl 16 . . . 4  |-  ( ph  ->  C  e.  CC )
2014, 17, 19rlim2 12966 . . 3  |-  ( ph  ->  ( ( z  e.  A  |->  B )  ~~> r  C  <->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  ( y  <_ 
z  ->  ( abs `  ( B  -  C
) )  <  x
) ) )
212, 20mpbid 210 . 2  |-  ( ph  ->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  (
y  <_  z  ->  ( abs `  ( B  -  C ) )  <  x ) )
22 breq2 4291 . . . . 5  |-  ( x  =  R  ->  (
( abs `  ( B  -  C )
)  <  x  <->  ( abs `  ( B  -  C
) )  <  R
) )
2322imbi2d 316 . . . 4  |-  ( x  =  R  ->  (
( y  <_  z  ->  ( abs `  ( B  -  C )
)  <  x )  <->  ( y  <_  z  ->  ( abs `  ( B  -  C ) )  <  R ) ) )
2423rexralbidv 2754 . . 3  |-  ( x  =  R  ->  ( E. y  e.  RR  A. z  e.  A  ( y  <_  z  ->  ( abs `  ( B  -  C ) )  <  x )  <->  E. y  e.  RR  A. z  e.  A  ( y  <_ 
z  ->  ( abs `  ( B  -  C
) )  <  R
) ) )
2524rspcv 3064 . 2  |-  ( R  e.  RR+  ->  ( A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  ( y  <_ 
z  ->  ( abs `  ( B  -  C
) )  <  x
)  ->  E. y  e.  RR  A. z  e.  A  ( y  <_ 
z  ->  ( abs `  ( B  -  C
) )  <  R
) ) )
261, 21, 25sylc 60 1  |-  ( ph  ->  E. y  e.  RR  A. z  e.  A  ( y  <_  z  ->  ( abs `  ( B  -  C ) )  <  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2710   E.wrex 2711    C_ wss 3323   class class class wbr 4287    e. cmpt 4345   dom cdm 4835   -->wf 5409   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273    < clt 9410    <_ cle 9411    - cmin 9587   RR+crp 10983   abscabs 12715    ~~> r crli 12955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-pm 7209  df-rlim 12959
This theorem is referenced by:  rlimi2  12984  rlimclim1  13015  rlimuni  13020  rlimcld2  13048  rlimcn1  13058  rlimcn2  13060  rlimo1  13086  o1rlimmul  13088  rlimno1  13123  xrlimcnp  22337  rlimcxp  22342  chtppilimlem2  22698  dchrisumlem3  22715
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