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Theorem rlimi 13315
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 13303 . . . . . . 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 2443 . . . . . . . . . 10  |-  ( z  e.  A  |->  B )  =  ( z  e.  A  |->  B )
76fmpt 6037 . . . . . . . . 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 5725 . . . . . . . 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 5708 . . . . . 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 6037 . . . . 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 13304 . . . . . 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 3524 . . . 4  |-  ( ph  ->  A  C_  RR )
18 rlimcl 13305 . . . . 5  |-  ( ( z  e.  A  |->  B )  ~~> r  C  ->  C  e.  CC )
192, 18syl 16 . . . 4  |-  ( ph  ->  C  e.  CC )
2014, 17, 19rlim2 13298 . . 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 4441 . . . . 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 2962 . . 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 3192 . 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 1383    e. wcel 1804   A.wral 2793   E.wrex 2794    C_ wss 3461   class class class wbr 4437    |-> cmpt 4495   dom cdm 4989   -->wf 5574   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494    < clt 9631    <_ cle 9632    - cmin 9810   RR+crp 11229   abscabs 13046    ~~> r crli 13287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-pm 7425  df-rlim 13291
This theorem is referenced by:  rlimi2  13316  rlimclim1  13347  rlimuni  13352  rlimcld2  13380  rlimcn1  13390  rlimcn2  13392  rlimo1  13418  o1rlimmul  13420  rlimno1  13455  xrlimcnp  23170  rlimcxp  23175  chtppilimlem2  23531  dchrisumlem3  23548
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