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Theorem rlimi 12307
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 12295 . . . . . . 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 2436 . . . . . . . . . 10  |-  ( z  e.  A  |->  B )  =  ( z  e.  A  |->  B )
76fmpt 5890 . . . . . . . . 9  |-  ( A. z  e.  A  B  e.  V  <->  ( z  e.  A  |->  B ) : A --> V )
85, 7sylib 189 . . . . . . . 8  |-  ( ph  ->  ( z  e.  A  |->  B ) : A --> V )
9 fdm 5595 . . . . . . . 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 5581 . . . . . 6  |-  ( ph  ->  ( ( z  e.  A  |->  B ) : dom  ( z  e.  A  |->  B ) --> CC  <->  ( z  e.  A  |->  B ) : A --> CC ) )
124, 11mpbid 202 . . . . 5  |-  ( ph  ->  ( z  e.  A  |->  B ) : A --> CC )
136fmpt 5890 . . . . 5  |-  ( A. z  e.  A  B  e.  CC  <->  ( z  e.  A  |->  B ) : A --> CC )
1412, 13sylibr 204 . . . 4  |-  ( ph  ->  A. z  e.  A  B  e.  CC )
15 rlimss 12296 . . . . . 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 3383 . . . 4  |-  ( ph  ->  A  C_  RR )
18 rlimcl 12297 . . . . 5  |-  ( ( z  e.  A  |->  B )  ~~> r  C  ->  C  e.  CC )
192, 18syl 16 . . . 4  |-  ( ph  ->  C  e.  CC )
2014, 17, 19rlim2 12290 . . 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 202 . 2  |-  ( ph  ->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  (
y  <_  z  ->  ( abs `  ( B  -  C ) )  <  x ) )
22 breq2 4216 . . . . 5  |-  ( x  =  R  ->  (
( abs `  ( B  -  C )
)  <  x  <->  ( abs `  ( B  -  C
) )  <  R
) )
2322imbi2d 308 . . . 4  |-  ( x  =  R  ->  (
( y  <_  z  ->  ( abs `  ( B  -  C )
)  <  x )  <->  ( y  <_  z  ->  ( abs `  ( B  -  C ) )  <  R ) ) )
2423rexralbidv 2749 . . 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 3048 . 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 58 1  |-  ( ph  ->  E. y  e.  RR  A. z  e.  A  ( y  <_  z  ->  ( abs `  ( B  -  C ) )  <  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    C_ wss 3320   class class class wbr 4212    e. cmpt 4266   dom cdm 4878   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989    < clt 9120    <_ cle 9121    - cmin 9291   RR+crp 10612   abscabs 12039    ~~> r crli 12279
This theorem is referenced by:  rlimi2  12308  rlimclim1  12339  rlimuni  12344  rlimcld2  12372  rlimcn1  12382  rlimcn2  12384  rlimo1  12410  o1rlimmul  12412  rlimno1  12447  xrlimcnp  20807  rlimcxp  20812  chtppilimlem2  21168  dchrisumlem3  21185
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-pm 7021  df-rlim 12283
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