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Theorem rlimeq 13068
Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 16-Sep-2014.)
Hypotheses
Ref Expression
rlimeq.1  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
rlimeq.2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
rlimeq.3  |-  ( ph  ->  D  e.  RR )
rlimeq.4  |-  ( (
ph  /\  ( x  e.  A  /\  D  <_  x ) )  ->  B  =  C )
Assertion
Ref Expression
rlimeq  |-  ( ph  ->  ( ( x  e.  A  |->  B )  ~~> r  E  <->  ( x  e.  A  |->  C )  ~~> r  E ) )
Distinct variable groups:    x, A    x, D    ph, x
Allowed substitution hints:    B( x)    C( x)    E( x)

Proof of Theorem rlimeq
StepHypRef Expression
1 rlimss 13001 . . 3  |-  ( ( x  e.  A  |->  B )  ~~> r  E  ->  dom  ( x  e.  A  |->  B )  C_  RR )
2 rlimeq.1 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
3 eqid 2443 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
42, 3fmptd 5888 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> CC )
5 fdm 5584 . . . . 5  |-  ( ( x  e.  A  |->  B ) : A --> CC  ->  dom  ( x  e.  A  |->  B )  =  A )
64, 5syl 16 . . . 4  |-  ( ph  ->  dom  ( x  e.  A  |->  B )  =  A )
76sseq1d 3404 . . 3  |-  ( ph  ->  ( dom  ( x  e.  A  |->  B ) 
C_  RR  <->  A  C_  RR ) )
81, 7syl5ib 219 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  B )  ~~> r  E  ->  A  C_  RR )
)
9 rlimss 13001 . . 3  |-  ( ( x  e.  A  |->  C )  ~~> r  E  ->  dom  ( x  e.  A  |->  C )  C_  RR )
10 rlimeq.2 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
11 eqid 2443 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1210, 11fmptd 5888 . . . . 5  |-  ( ph  ->  ( x  e.  A  |->  C ) : A --> CC )
13 fdm 5584 . . . . 5  |-  ( ( x  e.  A  |->  C ) : A --> CC  ->  dom  ( x  e.  A  |->  C )  =  A )
1412, 13syl 16 . . . 4  |-  ( ph  ->  dom  ( x  e.  A  |->  C )  =  A )
1514sseq1d 3404 . . 3  |-  ( ph  ->  ( dom  ( x  e.  A  |->  C ) 
C_  RR  <->  A  C_  RR ) )
169, 15syl5ib 219 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  C )  ~~> r  E  ->  A  C_  RR )
)
17 simpr 461 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  x  e.  ( A  i^i  ( D [,) +oo ) ) )
18 elin 3560 . . . . . . . . . . . . . 14  |-  ( x  e.  ( A  i^i  ( D [,) +oo )
)  <->  ( x  e.  A  /\  x  e.  ( D [,) +oo ) ) )
1917, 18sylib 196 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  ( x  e.  A  /\  x  e.  ( D [,) +oo ) ) )
2019simpld 459 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  x  e.  A )
2119simprd 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  x  e.  ( D [,) +oo )
)
22 rlimeq.3 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  D  e.  RR )
23 elicopnf 11406 . . . . . . . . . . . . . . . 16  |-  ( D  e.  RR  ->  (
x  e.  ( D [,) +oo )  <->  ( x  e.  RR  /\  D  <_  x ) ) )
2422, 23syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( x  e.  ( D [,) +oo )  <->  ( x  e.  RR  /\  D  <_  x ) ) )
2524biimpa 484 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( D [,) +oo )
)  ->  ( x  e.  RR  /\  D  <_  x ) )
2621, 25syldan 470 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  ( x  e.  RR  /\  D  <_  x ) )
2726simprd 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  D  <_  x )
2820, 27jca 532 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  ( x  e.  A  /\  D  <_  x ) )
29 rlimeq.4 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  D  <_  x ) )  ->  B  =  C )
3028, 29syldan 470 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A  i^i  ( D [,) +oo ) ) )  ->  B  =  C )
3130mpteq2dva 4399 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A  i^i  ( D [,) +oo ) ) 
|->  B )  =  ( x  e.  ( A  i^i  ( D [,) +oo ) )  |->  C ) )
32 inss1 3591 . . . . . . . . . 10  |-  ( A  i^i  ( D [,) +oo ) )  C_  A
33 resmpt 5177 . . . . . . . . . 10  |-  ( ( A  i^i  ( D [,) +oo ) ) 
C_  A  ->  (
( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )  =  ( x  e.  ( A  i^i  ( D [,) +oo ) ) 
|->  B ) )
3432, 33ax-mp 5 . . . . . . . . 9  |-  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )  =  ( x  e.  ( A  i^i  ( D [,) +oo ) ) 
|->  B )
35 resmpt 5177 . . . . . . . . . 10  |-  ( ( A  i^i  ( D [,) +oo ) ) 
C_  A  ->  (
( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) )  =  ( x  e.  ( A  i^i  ( D [,) +oo ) ) 
|->  C ) )
3632, 35ax-mp 5 . . . . . . . . 9  |-  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) )  =  ( x  e.  ( A  i^i  ( D [,) +oo ) ) 
|->  C )
3731, 34, 363eqtr4g 2500 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) ) )
38 resres 5144 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( A  i^i  ( D [,) +oo ) ) )
39 resres 5144 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  ( D [,) +oo ) ) )
4037, 38, 393eqtr4g 2500 . . . . . . 7  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) ) )
41 ssid 3396 . . . . . . . 8  |-  A  C_  A
42 resmpt 5177 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  B )  |`  A )  =  ( x  e.  A  |->  B ) )
43 reseq1 5125 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  B )  |`  A )  =  ( x  e.  A  |->  B )  -> 
( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( D [,) +oo ) ) )
4441, 42, 43mp2b 10 . . . . . . 7  |-  ( ( ( x  e.  A  |->  B )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  B )  |`  ( D [,) +oo ) )
45 resmpt 5177 . . . . . . . 8  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
46 reseq1 5125 . . . . . . . 8  |-  ( ( ( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C )  -> 
( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( D [,) +oo ) ) )
4741, 45, 46mp2b 10 . . . . . . 7  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  ( D [,) +oo ) )  =  ( ( x  e.  A  |->  C )  |`  ( D [,) +oo ) )
4840, 44, 473eqtr3g 2498 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  |->  B )  |`  ( D [,) +oo )
)  =  ( ( x  e.  A  |->  C )  |`  ( D [,) +oo ) ) )
4948breq1d 4323 . . . . 5  |-  ( ph  ->  ( ( ( x  e.  A  |->  B )  |`  ( D [,) +oo ) )  ~~> r  E  <->  ( ( x  e.  A  |->  C )  |`  ( D [,) +oo ) )  ~~> r  E ) )
5049adantr 465 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( ( x  e.  A  |->  B )  |`  ( D [,) +oo ) )  ~~> r  E  <->  ( ( x  e.  A  |->  C )  |`  ( D [,) +oo ) )  ~~> r  E ) )
514adantr 465 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  ( x  e.  A  |->  B ) : A --> CC )
52 simpr 461 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  A  C_  RR )
5322adantr 465 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  D  e.  RR )
5451, 52, 53rlimresb 13064 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  B )  ~~> r  E  <->  ( (
x  e.  A  |->  B )  |`  ( D [,) +oo ) )  ~~> r  E
) )
5512adantr 465 . . . . 5  |-  ( (
ph  /\  A  C_  RR )  ->  ( x  e.  A  |->  C ) : A --> CC )
5655, 52, 53rlimresb 13064 . . . 4  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  C )  ~~> r  E  <->  ( (
x  e.  A  |->  C )  |`  ( D [,) +oo ) )  ~~> r  E
) )
5750, 54, 563bitr4d 285 . . 3  |-  ( (
ph  /\  A  C_  RR )  ->  ( ( x  e.  A  |->  B )  ~~> r  E  <->  ( x  e.  A  |->  C )  ~~> r  E ) )
5857ex 434 . 2  |-  ( ph  ->  ( A  C_  RR  ->  ( ( x  e.  A  |->  B )  ~~> r  E  <->  ( x  e.  A  |->  C )  ~~> r  E ) ) )
598, 16, 58pm5.21ndd 354 1  |-  ( ph  ->  ( ( x  e.  A  |->  B )  ~~> r  E  <->  ( x  e.  A  |->  C )  ~~> r  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3348    C_ wss 3349   class class class wbr 4313    e. cmpt 4371   dom cdm 4861    |` cres 4863   -->wf 5435  (class class class)co 6112   CCcc 9301   RRcr 9302   +oocpnf 9436    <_ cle 9440   [,)cico 11323    ~~> r crli 12984
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 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-pre-lttri 9377  ax-pre-lttrn 9378
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-er 7122  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-ico 11327  df-rlim 12988
This theorem is referenced by: (None)
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