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Theorem rlimi 12034
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 12022 . . . . . . 7  |-  ( ( z  e.  A  |->  B )  ~~> r  C  -> 
( z  e.  A  |->  B ) : dom  ( z  e.  A  |->  B ) --> CC )
42, 3syl 15 . . . . . 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 2316 . . . . . . . . . 10  |-  ( z  e.  A  |->  B )  =  ( z  e.  A  |->  B )
76fmpt 5719 . . . . . . . . 9  |-  ( A. z  e.  A  B  e.  V  <->  ( z  e.  A  |->  B ) : A --> V )
85, 7sylib 188 . . . . . . . 8  |-  ( ph  ->  ( z  e.  A  |->  B ) : A --> V )
9 fdm 5431 . . . . . . . 8  |-  ( ( z  e.  A  |->  B ) : A --> V  ->  dom  ( z  e.  A  |->  B )  =  A )
108, 9syl 15 . . . . . . 7  |-  ( ph  ->  dom  ( z  e.  A  |->  B )  =  A )
1110feq2d 5417 . . . . . 6  |-  ( ph  ->  ( ( z  e.  A  |->  B ) : dom  ( z  e.  A  |->  B ) --> CC  <->  ( z  e.  A  |->  B ) : A --> CC ) )
124, 11mpbid 201 . . . . 5  |-  ( ph  ->  ( z  e.  A  |->  B ) : A --> CC )
136fmpt 5719 . . . . 5  |-  ( A. z  e.  A  B  e.  CC  <->  ( z  e.  A  |->  B ) : A --> CC )
1412, 13sylibr 203 . . . 4  |-  ( ph  ->  A. z  e.  A  B  e.  CC )
15 rlimss 12023 . . . . . 6  |-  ( ( z  e.  A  |->  B )  ~~> r  C  ->  dom  ( z  e.  A  |->  B )  C_  RR )
162, 15syl 15 . . . . 5  |-  ( ph  ->  dom  ( z  e.  A  |->  B )  C_  RR )
1710, 16eqsstr3d 3247 . . . 4  |-  ( ph  ->  A  C_  RR )
18 rlimcl 12024 . . . . 5  |-  ( ( z  e.  A  |->  B )  ~~> r  C  ->  C  e.  CC )
192, 18syl 15 . . . 4  |-  ( ph  ->  C  e.  CC )
2014, 17, 19rlim2 12017 . . 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 201 . 2  |-  ( ph  ->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  (
y  <_  z  ->  ( abs `  ( B  -  C ) )  <  x ) )
22 breq2 4064 . . . . 5  |-  ( x  =  R  ->  (
( abs `  ( B  -  C )
)  <  x  <->  ( abs `  ( B  -  C
) )  <  R
) )
2322imbi2d 307 . . . 4  |-  ( x  =  R  ->  (
( y  <_  z  ->  ( abs `  ( B  -  C )
)  <  x )  <->  ( y  <_  z  ->  ( abs `  ( B  -  C ) )  <  R ) ) )
2423rexralbidv 2621 . . 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 2914 . 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 56 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 1633    e. wcel 1701   A.wral 2577   E.wrex 2578    C_ wss 3186   class class class wbr 4060    e. cmpt 4114   dom cdm 4726   -->wf 5288   ` cfv 5292  (class class class)co 5900   CCcc 8780   RRcr 8781    < clt 8912    <_ cle 8913    - cmin 9082   RR+crp 10401   abscabs 11766    ~~> r crli 12006
This theorem is referenced by:  rlimi2  12035  rlimclim1  12066  rlimuni  12071  rlimcld2  12099  rlimcn1  12109  rlimcn2  12111  rlimo1  12137  o1rlimmul  12139  rlimno1  12174  xrlimcnp  20316  rlimcxp  20321  chtppilimlem2  20676  dchrisumlem3  20693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-pm 6818  df-rlim 12010
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