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Theorem rlimi 13112
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 13100 . . . . . . 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 2454 . . . . . . . . . 10  |-  ( z  e.  A  |->  B )  =  ( z  e.  A  |->  B )
76fmpt 5976 . . . . . . . . 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 5674 . . . . . . . 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 5658 . . . . . 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 5976 . . . . 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 13101 . . . . . 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 3502 . . . 4  |-  ( ph  ->  A  C_  RR )
18 rlimcl 13102 . . . . 5  |-  ( ( z  e.  A  |->  B )  ~~> r  C  ->  C  e.  CC )
192, 18syl 16 . . . 4  |-  ( ph  ->  C  e.  CC )
2014, 17, 19rlim2 13095 . . 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 4407 . . . . 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 2881 . . 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 3175 . 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 1370    e. wcel 1758   A.wral 2799   E.wrex 2800    C_ wss 3439   class class class wbr 4403    |-> cmpt 4461   dom cdm 4951   -->wf 5525   ` cfv 5529  (class class class)co 6203   CCcc 9394   RRcr 9395    < clt 9532    <_ cle 9533    - cmin 9709   RR+crp 11105   abscabs 12844    ~~> r crli 13084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-pm 7330  df-rlim 13088
This theorem is referenced by:  rlimi2  13113  rlimclim1  13144  rlimuni  13149  rlimcld2  13177  rlimcn1  13187  rlimcn2  13189  rlimo1  13215  o1rlimmul  13217  rlimno1  13252  xrlimcnp  22498  rlimcxp  22503  chtppilimlem2  22859  dchrisumlem3  22876
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